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ARITHMETIC  SIMPLIFIED, 


PREPARED  FOB  THE  USE  OP 


PRIMARY  SCHOOLS,  FEMALE  SEMINARIES, 


AND 


HIGH  SCHOOLS, 


IN   THREE    PARTS; 


ADAPTED  TO  CLASSES  OP  DIFFERENT  AGES,  AND  OF  DIFFERENT 
DEGREES  OF  ADVANCEMENT. 


BY  CATHARINE  E.  BEECHER, 

JLATE   PRINCIPAL   OF  THE    HARTFORD    FEUAI.E   SBHINART. 


HARTFORD : 
PUBLISHED  BY  D.'  F.  ROBINSON  &  CO. 

1832. 


Entered  according  to  Act  of  Congress,  in  the  year  1832,  by  D.  F. 
Robinson,  &  Co.  in  the  Clerk's  office  of  the  District  Court  of  Con- 
necticut. 


P.  CANTOELD.  PRINTER. 

HARTFORD. 


PREFACE.         lO  I 


15 


rcc 


The  public  have  this  claim  upon  any  author,  who  offers 
a  new  school  book,  that  such  a  work  shall  contain  some  es- 
sential advantages,  which  are  not  to  be  found,  in  any  othei 
work  already  in  use.  If  a  writer  cannot  sustain  such  a 
claim,  the  public  are  needlessly  taxed,  for  an  article  which 
is  not  wanted. 

It  therefore  seems  proper,  that  a  statement  should  be 
made  of  what  are  supposed  to  t)e,  the  peculiar  advantages 
and  improvements  in  this  work. 

The  writer,  for  several  years,  has  been  engaged  in  in- 
struction, and  has  either  used,  or  examined,  all  the  most 
popular  works  on  Arithmetic.  The  following  are  the  de- 
ficiencies, which  have  been  experienced,  and  which  it  is  the 
aim  of  this  work  to  supply.  It  should,  however,  be  pre- 
viously remarked,  that  all  these  difficulties  have  not  been 
experienced  in  every  work  of  the  kind,  heretofore  examined ; 
but  some  have  existed  in  one,  and  some  in  another,  and  no 
one  work,  yet  known  to  the  writer,  obviates  them  all. 

1.  The  first  difficulty,  for  which  a  remedy  is  here  at- 
tempted, originates  from  the  fact,  that  in 'every  school,  there 
is  such  a  variety  of  age,  intellect,  and  acquisition,  that  no 
(me  book  is  fitted  for  them  all.  If  a  work  is  found  adapted 
to  advanced  classes,  it  is  too  difficult  for  the  younger  and 
less  advanced.  If  it  is  fitted  to  these  last,  it  is  too  easy  for 
the  others. 

To  remedy  this,  the  following  work  is  divided  into  Three 
Parts.  The  First  Part  is  adapted  to  the  comprehension  of 
younff  children.  The  Second  Part  is  fitted  to  older  classes. 
The  Third  Part  completes  an  entire  system  of  Arithmetic, 
containincr  all  that  is  required  of  students  on  entering  col- 
lege. Th^e  whole  work  embraces  every  thing  of  any  con- 
sequence, that  can  be  found  in  the  most  complete  and  ex- 
tended works  ever  used,  and  yet  so  simplified  as  to  occupy 
much  less  space,  and  to  demand  much  less  labor.  There 
are  enough  subjects,  of  difficulty,  to  call  forth  mental  in- 
dustry and  effort,  without  allowing  those  which  are  plain 
to  remain  involved  in  needless  difficulties. 

2.  Another  difficulty  which  this  arrangement  remedies, 

1* 

865520 


Vi  PREFACE, 

arises  from  the  fact,  that  in  most  works  of  this  kind,  owing 
to  the  length  of  the  various  exercises  under  each  head,  the 
pupils  lose  the  general  principles  they  gain  in  one  part,  before 
they  reach  another.  Thus,  before  "Reduction  is  attained, 
the  principles  employed  in  Addition  and  Subtraction,  are 
partially  forgotten,  and  the  pupils  do  not  gain  a  clear  and 
general  view  of  the  whole  science. 

But  in  the  First  Part  of  this  work,  in  the  compass  of 
twenty  pages,  the  pupil  gains  all  the  fundamental  principles 
of  the  science,  which  in  the  succeeding  parts,  are  developed 
in  more  minute  particulars. 

To  aid  in  the  same  object,  and  to  secure  other  advanta- 
ges, a  new  method  of  classification  has  been  adopted.  The 
chiefbenefit  aimed  at,  by  this  new  arrangement,  is  to  sim- 
plify the  science,  by  leading  the  pupils  to  understand,  that 
all  the  various  exercises  of  Arithmetic  are  included  under 
the  same  general  principles  of  Addition,  Subtraction,  Mul- 
tiplication, Division,  and  Reduction. 

The  common  method  of  teaching  the  Simple  Rules  first, 
and  then  of  introducing  Vulgar  and  Decimal  Fractions, 
has  a  tendency  to  render  the  science  much  more  compli- 
cated and  perplexing.  Thus  the  pupil  is  first  taught  the 
process  of  Simple  Addition.  Then  follows  the  exercises  of 
the  other  simple  rules,  and  by  the  time  the  pupil  has  some- 
what forgotten  the  principles  of  Simple  Addition,  comes 
Compound  Addition,  which  seems  to  the  child  as  much  on 
a  new  principle,  as  Multiplication,  or  Division.  Then, 
after  another  interval,  comes  Decimal  Addition,  and  then 
the  Addition  of  Vulgar  Fractions. 

But  a  cliild  who  is  first  taught  the  system  of  numeration, 
as  including  whole  numbers  and  fractions,  and  the  nature  of 
each  of  these  modes  of  expressing  numbers,  can  immedi- 
ately commence  the  operation  of  Addition,  in  all  its  various 
particulars,  and  recognize  the  general  principle,  that  runs 
through  the  whole,  and  at  the  sam.e  time  the  peculiarity 
which  distinguishes  each. 

The  difficulties  arising  from  the  common  mode  of  ar- 
rangement, ai-e  particularly  felt,  when  the  processes  of 
multiplying  and  dividing  by  fractions,  are  introduced.  In  all 
previous  operations,  the  pupil  has  found  that  multiplication 
increases  a  number,  and  division  diminishes  it.  But  when 
fractions  are  introduced,  a  new  science  seems  to  commence, 
in  which  multiplication  lessens  and  division  increases  a  number,' 
and  all  heretofore  learned,  seems  to  be  contradicted  and 
undone. 
But  if,  at  the  commencement  of  the  science,  the  pupil  under- 


PREFACE.  VII 

Stands  the  peculiar  character  of  Fractions,  and  then  finds 
them  arranged  with  whole  numbers,  so  as  to  be  able  to 
compare  and  distinguish  the  same  general  principles  of  the 
various  exercises,  and  at  the  same  time  the  specific  differ- 
ence, the  perplexity  arising  from  an  apparent  multiplicity 
of  operations,  and  their  seemingly  contradictory  nature,  is 
avoidedj 

When  this  plan  was  first  attempted,  some  difficulty  was 
felt  from  the  necessity  of  the  operation  of  Division  in  the 
previous  operation  oi^  multiplying  by  a  fraction.  But  this  dif- 
ficulty has  been  obviated  by  having  the  First  Part  precede, 
in  which  Multiplication  and  Division  are  explained,  with- 
out entering  minutely  into  the  various  processes  of  Frac- 
tional Multiplication. 

It  will  be  found  tliat  pupils,  by  learning  the  Division  Ta- 
ble, and  the  First  Part,  can  perfonn  all  the  exercises  in 
Simple  and  Fractional  Multiplication,  without  any  other 
knowledge  of  the  i-ule  of  Division.  There  are  two  or  three 
exceptions,  liowever,  where  some  exercises  for  the  slate  are 
introduced,  where  the  rule  of  Division  must  beemployed. 
These  are  intended  for  older  pupils,  who  are  supposed  to 
understand  the  process  of  Division,  and  may  be  omitted  by 
new  beginners,  until  a  review. 

The  writer  has  herself  employed  this  method  of  classifi- 
cation, in  teaching,  and  it  has  been  used  by  the  teachers  in 
the  institution  under  Jier  care,  for  several  years ;  and  thus 
experience  lias  enforced  the  conviction,  which  at  first  was 
the  result  of  reasoning,  that  this  mode  of  classification  will 
better  secure  the  benefits  sought  for,  in  all  attempts  at  gen- 
eralization. It  certainly  attains  advantages,  and  avoids 
difficulties,  much  more  than  the  common  method. 

3.  Another  difficulty  experienced  in  using  same  of  the 
most  popular  works  of  this  kind,  has  arisen  from  the  fact, 
that  the  mental  and  loritten  exercises  2iave  been  entirely 
separate ;  in  some  cases  being  placed  in  different  books. 
Thus  the  pupil,  after  completing  an  Aritlunetic  designed 
for  mental  exercise  alone,  will  often  be  found  repeatino-  ex- 
actly the  same  processes  in  written  Arithmetic,  witliout 
recognizing  the  principles,  which,  in  mental  operations, 
have  been  constantly  employed.  To  remedy  this,  both 
mental  and  written  exercises  are  placed  together  under 
evei'y  general  rvale. 

4.  Another  defect  in  teaching  this  science,  has  arisen 
from  a  want  of  some  method  of  stating  and  explaining  the 
rationale,  of  each  arithmetical  process.  In  many  Arithme- 
tics, a  mechanical  method  is  presented,  of  performing  cer- 


VIU  PREFACE. 

tain  operations  according  to  rule,  without  any  exhibition  of 
the  reason  for  such  operations.  Thus,  in  Subtraction,  tchy 
one  is  carried,  and  ten  borrowed ;  or,  in  Multiphcation,  why 
the  figures  are  placed  in  a  certain  method  ;  or,  in  Division, 
why  multiphcation  and  subtraction  are  performed,  is  never 
explained  or  illustrated.  To  a  child,  they  are  a  sort  of  ca- 
balistical  process,  which  he  finds  will  bring  the  right  answer, 
and  this  is  all  he  can  know  from  any  thing  he  gains  from 
the  book.  To  remedy  this,  in  the  following  work,  every 
rule  is  accompanied  by  a  full  explanation  of  the  reason,  for 
each  process  employed.  In  the  mental  operations  also,  a 
proper  mode  of  stating  each  process  is  given.  The  defini- 
tions, rules,  and  explanations,  will  be  found  to  iDe  more  simple 
and  concise  than  in  many  works  of  this  kind,  and  perliaps 
may  be  considered  as  improvements. 

As  the  writer  has  been  in  a  situation,  in  which  she  has 
had  to  employ  various  teachers,  of  diflferent  qualifications, 
it  has  been  one  great  aim  to  furnish  a  work,  by  which  neie 
and  inexperienced  teachers  could  avail  themselves  of  the  ex- 
perience of  others.  This  work  has  been  maturing  for 
several  years,  and  the  results  of  the  experience  of  several 
able  and  ingenious  teachers  employed  by  the  writer,  have 
been  added  to  her  own.  It  is  believed  that  any  teachers  with 
common  talents  and  industry,  can,  with  the  aid  of  this 
work,  do  all  for  their  pupils  in  this  science,  which  needs  to 
be  done,  in  order  to  make  them  thorough  and  expert  pro- 
ficients. 

For  the  purpose  of  perfecting  such  a  work  as  this,  and 
to  make  a  fair  trial  of  the  several  improvements  contem- 
plated, a  small  work  on  this  plan  was  printed  some  years 
ago,  for  the  use  of  the  pupils  of  the  writer.  But  as  it  was 
intended  for  an  experiment,  and  was  necessarily  very  im- 
perfect and  incomplete,  it  was  never  published.  Yet,  as  in 
several  cases,  those  who  have  been  teachers  and  pupils  in 
this  institution,  have  introduced  it  into  their  schools,  it  may 
be  proper  to  add,  that  this  work  is  very  different  from  the 
former,  being  much  easier,  much  more  extensive  and  com- 
plete, and  improved  in  several  respects  which  it  is  unne- 
cessarj^  to  mention. 

The  writer  does  not  lay  claim  to  any  great  originality, 
in  these  various  particulars,  but  has  aimed  to  unite  in  one 
work  the  various  excellencies,  which  might  be  otherwise 
scattered  among  a  variety. 

Hartford  Female  Seminary,  Jan.  1,  1S32. 


TO    TEACHERS. 


It  is  veiy  desirable  that  new  beginners  should  review  the 
First  Part,  till  it  is  vei-y  thoroughly  understood.  It  will  save 
much  trouble  to  both  teacher  and  pupil. 

It  will  be  found  advantageous,  to  require  older  pupils  te 
study  the  First  Past,  previous  to  commencing  the  Second  ; 
for  though  some  of  the  exercises  are  very  simple,  there  are 
some  important  explanations  and  illustrations,  not  found  in 
the  Second  Part. 

It  is  very  desirable  that  pupils  should  become  thorough 
and  expert  in  Numeration,  especially  in  Decimal  Numera- 
tion, before  taking  the  next  lessons,  and  one  or  two  reviews 
are  recommended  previous  to  proceeding. 

In  Compound  Addition  and  Fractional  Multiplication,  if  the 
pupil  has  never  practised  Simple  Division,  omit  those  exer- 
cises which  require  this  rule,  till  a  review. 

The  Second  Part  should  be  reviewed,  before  commencing 
the  Third  Part. 

If  any  teachers  have  a  preference  for  the  common 
method  of  classification,  it  is  very  easy  to  direct  the  pupils 
to  learn  the  Shnple  Rules  first.  But  every  pupil  will  find  it 
advantageous  at  least  to  review  on  theplanofan'angement 
adopted  in  this  work. 

When  young  beginners  take  the  Second  Part,  it  is  re- 
commended, that  they  take  the  easiest  exercises,  and  re- 
serve the  more  difficult,  till  a  review. 


ERRATA. 

N.  B.  Pupils  are  requested  to  make  these  corrections  with  a  pen,  before 
prooeeding  to  study  the  book. 

Page  39,  line  13  from  tire  bottom,  for  ascending  read  descending.  P.  53,  line 
12  from  the  bottom,  for  orders  read  periods.  P.  82,  line  14  from  the  bottom, 
change  the  signs  from  multiplication  to  addition.  P.  92,  line  10,  for  gr. 
read  grs.  P.  98,  line  8  from  the  bottom,  for  2  read  8.  P.  110,  line  1  for  10, 
read  18  ;  in  the  answer  to  the  fifth  sum,  for  5  read  11  ;  the  answer  to  the 
seventh  sum  should  be  256  yds.  ;  line  8  from  the  bottom,  for  grs.  read  roods. 
P.  118,  line  6  from  the  bottom,  for  A^  read  12.     P.  169,  lines  6  and  14,  for  3  read  2 

'  1  2  'in* 

P.  173,  line  5,  for  lb.  read  £.  P.  200,  at  the  end  of  line  3  from  the  bottom,  insert 
at  6 per  cent.  P.  201,  at  line  15,  insert  in  1  yr.  4  mo.  P.  233,  lines  4  and  5  from 
the  bottom,  for  seconds  and  thirds,  read  twelfths  and  seconds. 

In  many  places  in  the  first  and  second  parts  of  the  work,  an  error  will  be  per- 
ceived in  the  manner  of  expressing  decimals,  tens  and  hundreds  being  used  in- 
stead of  «cntA«  and  hundredths  ;  thus,  ten$  of  thousandths,  instead  of  tenMs  of 
thousandths,  or  ten  thousandths. 


INDEX. 


ARITHMETICAL  TABLES. 


Addition  Table,  .... 

Subtraction  Table,  .  .  .  . 

Multiplication  Table, 

Division  Table       .  -  -  .  - 

Table  of  Weights  and  Measures, 

Table  of  Foreign  Coins  in  Federal  Money, 

Table  of  Scripture  Weights,  Measures,  and  Coins, 


ARITHMETIC— PART  FIRST. 


Addition, 

Subtraction, 

Multiplication, 

Division, 

Reduction, 


Page  13 
13 
14 
15 
16 
19 
19 


25 
26 

28 
31 
37 


PART  SECOND. 

Numeration.  * 

Numeration  of  Whole  Numbers, 
Numeration  of  Vulgar  Fractions, 
Decimal  Numeration, 

Addition, 

Simple  Addition, 

Decimal  Addition, 

Compound  Addition, 

Addition  of  Vulgar  Fractions, 

Subtraction. 

Simple  Subtraction, 

Decimal  Subtraction, 

Compound  Subtraction, 

Subtraction  of  Vulgar  Fractions, 

Multiplication. 

Simple  Multiplication, 

Decimal  Multiplication, 

Compound  Multiplication, 

Multiplication  of  Vulgar  Fractions, 

Division, 

Simple  Division, 

Compound  Division, 


42 
55 
58 
66 
67 
73 
78 
82 

83 
86 
91 
93 

94 
103 
109 
111 
123 
124 
132 


INDEX.  XI 

Division  of  Vulgar  Fractions,        -            -            -            -  135 

Decimal  Division,               ....            -  144 

Reduction. 

Reduction  Ascending  and  Descending,      -            -            -  154 

Reduction  of  Fractions  to  Whole  Numbers,              •          -  158 

Reduction  of  Whole  Numbers  to  Fractions,             -             -  159 

Reductionof  Vulgar  to  Decimal  Fractions,              -             -  160 

Reduction  of  Fractions  to  a  Common  Denominator,             -  162 

Reduction  of  Fractions  to  the  Lowest  Terms,         -             -  167 

Reduction  of  Fractions  from  one  order  to  another,             -  170 
Reduction  of  Fractions  of  one  order,  to  Units  of  another  order,    17X 

Reduction  of  Units  of  one  order  to  Fractions  of  another  order,  172 

Reduction  of  a  Compound  Number  to  a  Decimal,                 -  173 

Reduction  of  a  Decimal  to  Units  of  Compound  Orders,      -  175 

Reduction  of  Currencies,                 ...             -  176 

Reduction  of  Different  Currencies  to  Federal  Money,        -  179 

Reductionof  Federal  Money  to  Different  Currencies,         -  l80 

Reduction  from  one  Currency  to  another,                -            -  181 

PART  THIRD. 

Numeration. 

Roman  Numeration,            .....  184 

Other  methods  of  Numeration,       ....  185 

Common,  Vulgar,  and  Decimal  Numeration,          -            -  186 

Addition. 

Simple,  Vulgar,  and  Decimal  Addition,       -            -            -  187 

Subtraction. 

Simple,  Vulgar,  and  Decimal  Subtraction,             .            -  188 

Multiplication. 

Simple,  Vulgar,  Compound,  and  Decimal  Multiplication,  189 

Division. 

Simple,  Vulgar,  Compound,  and  Decimal  Division,            -  191 

Exercises  in  Reduction      .....  193 

Interest,                 -             -            .            .            .            -  l94 

Simple  Interest,                  .....  196 

To  find  the  Interest  on  Sterling  Money,     -            •            -  200 

Various  Exercises  in  Interest,        ....  200 

Endorsements,        ......  202 

First  Method,         ......  203 

Massachusetts  Rule,          .....  204 

Connecticut  Rule,                .....  205 

Compound  Interest,            .....  208 

Discount,                ......  210 

Stock,  Insurance,  Commission,  Loss  and  Gain,  Duties,    .  210 
EauATioN  of  Payments,                 .           .            -           .214 

Ratio,        ....            .            .            -  215 


XII 


INDEX. 


Proportion,  ......  216 

Simple  Rule  of  Three;  or  Simple  Proportion,       .  .         218 

Double  Rule  of  Three  ;  or  Compound  Proportion,  .  221 

Fellowship,  ......  225 

Alligation,  -  .  .  .  .  .  228 

DrjoDJEciMALs,         ......  232 

Involution,  ......  235 

Evolution,  ......  237 

Extraction  of  the  Square  Root,      ....  239 

Extraction  of  the  Cube  Root,  ....  243 

Arithmetical  Progression,  ....  248 

Geometrical  Progression,  ....  252 

Annuities,  ......  254 

Permutation,  .  -  .  .    '        -  .  259 

Miscellaneous  Examples,  ....  260 

Position,  Ex.  40—50.     Barter,  Ex.  51—58. 
To  find  the  Area  of  a  Square,  Ex.  69 — 71. 

„  „     of  a  Parallelogram,  Ex.  72—74. 

„     of  a  Triangle,         Ex.  75,  76. 
To  find  the  Solid  Contents  of  a  Cube,  Ex.  77 — 79. 
To  find  the  Circumference,  Diameter  and  Area  of  a  Circle, 

Ex.  80—84. 
To  find  the  Area  of  a  Globe  or  Ball,  Ex.  85. 
To  fiind  the  Solid  Contents  of  a  Globe  or  Ball,  Ex.  86. 

„  „  „        of  a  Cylinder,  Ex.  87. 

„  „       of  a  Pyramid,  Ex.  88,  89- 

FoRMs  OF  Notes,  Receipts,  and  Orders,   .  -  -  269 

Book-Keeping,       ......         272 


ARITHMETICAL  TABLES. 


ADDITION  TABLE. 


■Q 

and 

0 

are    2 

4 

and 

0 

are    46 

and 

0 

are  6 

8 

and 

0 

are  8 

s 

1 

3,4 

1 

56 

I 

7 

8 

1 

9 

3 

2 

4!4 

2 

6'6 

2 

8 

8 

2 

10 

5 

3 

514 

3 

716 

3 

9 

8 

3 

11 

2 

4 

6 

4 

4 

8l6 

4 

10 

8 

4 

12 

2 

5 

7 

4 

5 

96 

5 

11 

8 

5 

IS 

2 

6 

8 

4 

r> 

10(i 

6 

•  12 

8 

6 

14 

2 

7 

9 

4 

7 

Iljfi 

7 

13 

8 

7 

15 

2 

8 

10 

4 

8 

126 

8 

14 

8 

8 

16 

2 

9 

11 

4 

9 

1316 

9 

15 

8 

9 

17 

3 

and 

0 

are    3 

5 

and 

0 

are    5;7 

and 

0 

are  7 

9 

and 

0 

are  9 

3 

1 

4 

5 

1 

67 

I 

8 

9 

1 

10 

3 

2 

5 

5 

2 

77 

2 

9 

9 

2 

11 

3 

3 

6 

5 

3 

87 

3 

10 

9 

3 

12 

3 

4 

7 

5 

4 

9|7 

4 

I] 

9 

4 

13 

3 

5 

8 

5 

5 

10  7 

5 

12 

9 

5 

14 

3 

6 

9 

5 

G 

117 

6 

13 

9 

6 

15 

3 

7 

10 

5 

7 

12J7 

7 

14 

9 

7 

16 

3 

8 

11 

5 

8 

137 

8 

15 

9 

8 

17 

3 

9 

12 

5 

9 

147 

9 

16 

9 

9 

18 

SUBTRACTION  TABLE. 


from 

1 

leaves 

0 

from 

4    leaves    07 

from 

7 

leaves 

0 

2 

] 

5 

1|7 

8 

1 

3 

2 

6 

2|7 

9 

2 

4 

3 

7 

3,7 

10 

3 

5 

4 

8 

47 

11 

4 

6 

5 

9 

57 

12 

5 

7 

6 

10 

6(7 

13 

6 

8 

7 

11 

7:7 

14 

7 

9 

8 

12 

8 

7 

15 

8 

10 

9 

4 

13 

9 

7 

16 

9 

2 

from 

2 

leaves 

0 

5 

from 

5    leaves    0 

8 

from 

8 

leaves 

0 

2 

3 

1 

5 

6 

1 

8 

9 

1 

2 

4 

2 

5 

7 

28 

10 

2 

2 

5 

3 

5 

8 

38 

11 

3 

2 

6 

4 

5 

9 

48 

12 

4 

2 

7 

5 

5 

10 

58 

13 

5 

2 

8 

6 

5 

11 

68 

14 

6 

2 

9 

7 

5 

12 

7|8 

15 

7 

2 

10 

8 

5 

13 

sis 

16 

8 

2 

11 

9 

5 

14 

9 

8 

17 

9 

3 

from 

3 

leaves 

0 

6 

from 

6 

0 

9 

frcan 

9 

leaves 

0 

3 

4 

1 

6 

7 

I 

9 

10 

1 

3 

5 

2 

6 

8 

2 

9 

11 

2 

3 

6 

3 

6 

9 

3 

9 

12 

3 

3 

7 

4 

6 

10 

4 

9 

13 

4 

3 

8 

5 

6 

11 

5 

9 

14 

5 

3 

9 

6 

6 

12 

6 

9 

15 

6 

3 

10 

7 

6 

13 

7 

9 

16 

7 

3 

11 

8 

6 

14 

8 

9 

17 

8 

3 

12 

9 

6 

15 

9 

9 

18 

9 

14 


MULTIPLICATION    TABLE. 


MULTIPLICATION  TABLE. 


2  times  0  ' 

ire    0 

5  times  0  are     0] 

3  times  0    are    0 

11  t 

mes  0 

are     0 

2 

X  1 

=  2 

5 

X    1 

=  5 

8  X 

1  =  8 

11 

X    1  = 

=    11 

2 

2 

4 

5 

2 

10 

8 

2       16 

11 

2 

22 

2 

3 

6 

5 

3 

15 

8 

3       24 

11 

3 

33 

2 

4 

8 

5 

4 

20 

8 

4       32 

U 

4 

44 

2 

5 

10 

5 

5 

25 

8 

5       40 

11 

5 

55 

2 

6 

12 

5 

6 

30 

8 

6       48 

11 

6 

66 

2 

7 

14 

5 

7 

35 

8 

7       56 

11 

7 

77 

2 

8 

16 

5 

8 

40 

8 

8       64 

11 

8 

88 

2 

9 

18 

5 

9 

45 

8 

9       72 

11 

9 

99 

2 

10 

20 

5 

10 

50  8 

10       80 

11 

10 

110 

2 

11 

22 

5 

11 

558 

11       88 

11 

11 

121 

2 

12 

24 

5 

12 

G0j8 

12       96 

11 

12 

132 

3  times  0 

are   0 

6  times  0 

are    0 

9  times  0    are    0 

12 

imes  0 

are     0 

3 

X    I 

=  3 

6 

X    1 

=:     6 

9x1  =  9 

12 

X    1 

=  12 

3 

2 

6 

6 

2 

12 

9 

2         IH 

12 

2 

24 

3 

3 

9 

6 

3 

18 

9 

3       27 

12 

3 

36 

3 

4 

12 

6 

4 

24 

9 

4       36 

12 

4 

48 

3 

5 

15 

6 

5 

30 

9 

5       45 

12 

5 

60 

3 

G 

18 

6 

6 

36)9 

6       54 

12 

6 

72 

3 

7 

21 

6 

7 

42 

9 

7       63 

12 

7 

84 

3 

8 

24 

6 

8 

48 

9 

8       72 

12 

8 

96 

3 

9 

27 

6 

9 

54 

9 

9       81 

12 

9 

108 

3 

10 

30 

G 

10 

60 

9 

10       90 

12 

10 

120 

8 

11 

33 

6 

11 

66 

9 

11       99 

12 

11 

132 

3 

12 

36 

6 

12 

72 

9 

12     108 

12 

12 

144 

4  times  0 

are     0 

7t 

imes  0 

are    0 

lOti 

mesO  ai-e     Q 

13  times  0 

are     0 

4 

X    1 

=   4 

7 

X    1 

=  7 

10 

X  1=10 

13 

X  1  = 

=    13 

4 

2 

8 

7 

2 

14 

10 

2      20 

13 

2 

26 

4 

3 

12 

7 

3 

21 

10 

3      30 

13 

3 

39 

4 

4 

16 

7 

4 

28 

10 

4      40 

13 

4 

52 

4 

5 

20 

7 

5 

35 

10 

5      50 

13 

5 

65 

4 

6 

24 

r 

6 

42 

10 

6      60 

13 

6 

78 

4 

7 

28 

7 

7 

49 

10 

7      70 

13 

7 

91 

4 

8 

32 

7 

8 

56 

10 

8      80 

13 

8 

104 

4 

9 

36 

7 

9 

63 

10 

9      90 

13 

9 

117 

4 

10 

40 

7 

10 

70 

10 

10    100 

13 

10 

130 

4 

11 

44 

7 

11 

77 

10 

11     110 

13 

11 

143 

4 

12 

48 

7 

12 

84 

10 

12    120 

13 

12 

156 

DIVISION    TABLE. 
DIVISION  TABLE. 


15 


2  in     2 

11 

6  in     6 

n 

10  in 

10 

11 

2         4 

2 

6       12 

2 

10 

20 

2 

2         0 

3 

0       18 

3 

10 

30 

3 

2         8 

4 

j^_ 

G       24 

4 

-     10 

40 

4 

«■ 

2       10 

5}  i   1 

6       30 

5 

i      10 

50 

5 

<0 

2       12 

6 

CO 

6      36 

6 

CO 

10 

60 

6 

00 

2       14 

7 

6      42 

7 

10 

70 

7 

2       16 

8 

6       48 

8 

10 

80 

8 

2       18 

9. 

6       54 

9, 

10 

90 

9j 

Sin     3 

11 

7  in     7 

11 

11  in 

11 

11 

3         6 

2 

7       14 

2 

11 

22 

2 

3         9 

3 

7       21 

3 

11 

33 

3 

3       12 

4 

- 

7       28 

4 

^ 

11 

44 

4 

r. 

3       15 

5 

^1 

7       35 

5 

11 

55 

5 

\i 

3       18 

6 

f 

7       42 

6 

CO 

11 

66 

6 

CO 

3       21 

7 

7       49 

7 

11 

77 

7 

3      24 

8 

7       56 

8 

11 

88 

8 

3       27 

9, 

7       63 

9J 

11 

99 

9, 

4  in     4 

11 

8   n    8 

1^ 

12  in 

12 

1^ 

4         8 

2 

8       16 

2 

12 

24 

2 

4       12 

3 

8       24 

3 

12 

36 

3 

4       16 

4 

r- 

S       32 

4 

r» 

12 

48 

4 

rt. 

4       20 

5 

•e' 

8      40 

5 

>i 

12 

60 

5 

J" 

4       24 

6 

01 

8       48 

6 

CO 

12 

72 

6 

CQ 

4       28 

7 

8       56 

7 

12 

84 

7 

4       32 

8 

8       64 

8 

12 

96 

8 

4       36 

9, 

8       72 

9, 

1 

12 

108 

9 

5  in     5 

r 

9  in    9 

1~ 

1 

13  in 

13 

1- 

5       10 

2 

9       18 

2 

13 

26 

2 

5       15 

3 

9       27 

3 

13 

89 

3 

5       20 

4 

r» 

9       36 

4 

r* 

13 

52 

4 

«- 

5       25 

5 

CO 

9       45 

5 

.  3" 

13 

65 

5 

.  3' 

CD 

5       30 

6 

m 

9       54 

6 

M 

13 

78 

6 

CO 

5       35 

7 

9       63 

7 

13 

91 

7 

5       40 

8 

9      72 

8 

13 

104 

8 

5       45 

9^ 

9       81 

9 

13 

117 

9, 

16 


WEIGHTS  AND  MEASURES. 


1.   Troy  Weight. 

24  grains  (gr.)  make         1  periuy-weight,  marked  pwt^ 

20  penny-weights,              1  ounce,  oz. 

12  ounces,                         1  pound,  lb. 

2.  Avoirdupois  Weight. 

16  drams  {dr.)  make        1  ounce,  oz. 

16  ounces,                          1  pound,  lb. 

28  pounds,  1  quarter  of  a  hundred  weight,  qr. 

4  quarters,                        1  hundred  weight,  cwt. 

20  hundred  weight,            1  ton.  T. 
By  this  weight  are  weighed  all  coarse  and  drossy  goods, 
grocery  wares,  and  all  metals  except  gold  and  silver. 

3.  Apothecaries  Weight. 

20  grains  (gr.)  make            1  scruple,  9 

3  scruples,                           1  dram,  3 

8  drams,                               1  ounce,  g 

12  ounces,                              1  pound,  ft 

Apothecaries  use  this  weight  in  compounding  their 

medicines. 


4.  Cloth  Measure. 

4  nails  (na.)  make               1  quarter  of  a  yard, 

qr. 

4  quarters,                           1  yard, 

yd. 

3  quarters,                           1  Ell  Flemish,. 

E.Fl. 

5  quarters,                           1  Ell  English, 

E.E. 

6  quarters,                          1  Ell  French, 

E.Fr. 

5.  Dry  Measure. 
2  pints  (pi.)  make  1  quart,  qt. 

8  quarts,  1  peck,  pk. 

4  pecks,  1  bushel,  hu. 

This  measure  is  applied  to  grain,  beans,  flax-seed,  salt, 
oats,  oysters,  coal,  &;c. 


pt. 

qt. 

gal. 

bL 


6. 

Wine  Measure. 

4  gills  {gi.) 

make 

1  pint, 

2  pints, 

1  quart. 

4  quarts, 

1  gallon. 

31^  gallons, 

1  barrel, 

WEIGHTS  AND  MEASURES.  17 

42  gallons,  1  tierce,  tier. 

63  gallons,  1  hogshead,  hhd. 

2  hogsheads,  1  pipe,  p. 

2  pipes,  1  tun,  T. 
AH  brandies,  spirits,  mead,  vinegar,  oil,  &c.  are  meas- 
ured by  wine  measure.     Note. — 231   solid  inches,  make 
a  gallon. 

7.  Long  Measure. 

3  barley  corns  (&.  c.)  make  1  inch,  marked  in. 
12  inches,                                     1  foot,                            ft. 

3  feet,  1  yard,  yd. 

5|  yards,  1  rod,  pole,  or  perch,  rd. 

40  rods,  1  furlong,  fur. 

8  furlongs,  1  mile,  m. 

3  miles,  1  league,  lea. 

69^  statute  miles,  I  degree,  on  the  earth. 

'360  degrees,  the  circumference  of  the  earth. 

The  use  of  long  measure  is  to  measure  the  distance  of 
places,  or  any  other  thing,  where  length  is  considered, 
without  regard  to  breadth. 

N.  B.  In  measuring  the  height  of  horses,  4  inches  make 
1  hand.  In  measuring  depths,  six  feet  make  one  fathom  or 
French  toise.  Distances  are  measured  by  a  chain,  four 
rods  long,  containing  one  hundred  links. 

8.  Land,  or  Square  Measure. 

144  square  inches  make  1  square  foot. 

9  square  feet,  1  square  yard. 

30|  square  vards,  or  >  ,  , 

«-r.-il  r    .  >  1  square  rod. 

272;^  square  leet,  ^  ^ 

40  square  rods,  1  square  rood. 

4  square  roods,  1  square  acre. 

640  square  acres,  1  square  mile. 

Note. — In  measuring  land,  a  chain,  called  Gunter's  chain,  4  rods 
in  length,  is  used.  It  is  divided  into  100  links.  Of  course,  25  links 
make  a  rod,  and  25  times  25=625  square  links  make  a  square  rod. 
In  4  rods,  there  are  792  inches.     Of  course,  1  link  is  7  -|-|_ 

9.  Solid,  or  Cubic  Measure. 
1728  solid  inches  make  1  solid  foot. 


2* 


18  WEIGHTS  AND  MEASURES. 

40  feet  of  round  timber,  or  >  ,  ^  ,      , 

cnr    »    ru        4k  t  1  ton  or  load. 

oU  leet  oi  hewn  timber,        ^ 

128  solid  feet  or  8  feet  long,  i  ^  cord  of  wood, 

4  wide,  and  4  high,  ^ 

All  solids,  or  things  that  have  length,  breadth  and  depth, 
are  measured  by  this  measure.  N.  B.  The  wine  gallon  con- 
tains 231  solid  or  cubic  inches,  and  the  beer  gallon,  282.. 
A  bushel  contains  2150,42  solid  inches. 

10.   Time. 
60  seconds  (<S'.)  make         1  minute,  marked  M. 

60  minutes,  1  hour,  h. 

24  hours,  1  day,  d. 

7  days,  1  week,  w. 

4  weeks,  1  month,  mo. 

13  months,  1  day  and  6  hours,  I  Julian  year,  yr. 

Thirty  days  hath  September,  April,  June,  and  Novem- 
ber, February  tv/enty-eight  alone,  all  the  rest  have  thirty, 
one. 

N.  B.  In  bissextile  or  leap-year,  February  hath  29 
days. 

II.  Circular  Motion. 
60  seconds  (")  make  1  minute,  '' 

60  minutes,  1  degree,  ° 

30  degrees,  1  sign,  S, 

12  signs,  or  360  degrees,  the  whole  great  circle  of  the 
Zodiac. 

12  units  make  A  Dozen. 

12  dozen  A  Gross. 

144  dozen  A  Great  Gross* 

20  units  A  Score. 

24  sheets  of  paper  A  Quire. 

20  quires  A  Ream. 


VALUE    OF   FOREIGN    COINS. 


19 


Value  of  Foreign  Corns  in  Federal  Money- 

RixDollar  of  Austria,  0.778— 

Rix  Dollar  of  Denmark  ^             ..» 

and  Switzerland,  J 

RixDollai*  of  Sweden,  1.037 

Rix  Dollar  of  Prussia,  0.778— 

F'lorin,                   "  0.259+ 

Ducat  of  Sweden  and   i  omd 

Prussia,  J 

Piaster  of  ex,  of  Spain,  0.80 

Ducat  of  ex,«       "  l.ioa— 

Stiver           of  Holland,  0.019-|- 

Cniilder  or  Florin,  "  0.388 

RixDollar,            "  0.970 

Ducat,                    "  2.079 

Gold  Ducat,          "  8.000 

Ducat  of  Denmark,  8.833+ 

Ruble,  of  Russia,  1.000 

Ziuvonitz,     "  2.000 

Tale,  of  China,  1.480 

Pagoda,  of  India,  1.840 

Rupen,  of  Bengal,  0.500 

Xeriir,  of  Turkey,  2.222 

*  Those  denominations  which  have  the  asterisk,  (as  the  Pistole  of  France, 
and  the  Milre  of  Portugal,)  are  merely  nominal;  that  is,  they  arc  represented 

by  no  real  coin.     In  this  respect,  they  are  like  the  Mill  in  Federal  Money. 


Shilling  Sterling, 

$0,222  ' 

Crown  5s. 

I.IU 

Sovereign,  (a  gold  coin,  =  £,)   4.414 

Guinea,  (21s.  nearly  out  of 

i    4.666 

use  m  England,) 

Livre              of  France, 

0.185+ 

Franc                     " 

0.1875— 

Pistole*  10  livres" 

1.852— 

Louis  d'or,             " 

4,444+ 

Five  franc  [liece,   " 

0.937 

Realof  Plate,  of  Spain, 

0.100 

RealofVellon,      " 

0.050 

Pistole,                  " 

3.60 

Dollau                  " 

1.00 

Re.  of  Portugal, 

80.0012+ 

Testoon,     " 

0.125 

Milre,*       " 

1.250 

Moidore,    " 

6.000 

Joanese,     " 

8.000 

Marc  Banco  of  Haniburgli, 

0.333+ 

Pistole  of  Italy, 

3.200 

A  TABLE  OP  SCRIPTURE  WEIGHTS,  MEASURES,  AND  MONEY. 


MEASURES    OF 


A  Cubit, 

A  Span,  half  cubit, 

A  Hand  breadth, 

A  Finger, 

A  Fathom, 

Ezekiel's  reed, 

The  measuring  line. 

Sabbath  day's  journey. 
Eastern  mile, 
Stadium,  or  Furlong, 
Day's  journey, 


LENGTH. 

feet. 
1 

0 

0 

0 

7 

10 

145 

miles,    furlongs. 

0  5 

1  3 
0  1 

33  1 


inches. 
9,88 

10,94 
3,68 
0,91 
3,55 

11,32 

11,04 

rods.    feet. 


21 
2 
4 

12 


H 
3 

3 

6 


20 


SCRIPTURE  WEIGHTS  AND  MEASURES. 


MEASURE    OF   LIOUIDS. 


gall 

pints,     sol.  inch. 

The  Homer  or  Cor. 

75 

5 

7,6 

The  Bath,         .         .         . 

7 

4 

15,2 

The  Hin, 

1 

2 

2,5 

The  Log, 

0 

0 

24,3 

The  Firkin,      . 

0 

7 

4,9 

MEASURE    OF 

THINGS. 

bushels,    pecks. 

pints. 

The  Homer, 

8 

0 

1,6 

The  Lethech, 

4 

0 

0,8 

TheEphah, 

0 

3 

3,4 

The  Seah, 

0 

1 

1,1 

The  Otner, 

0 

0 

5,1 

The  Cab, 

0 

0 

2,9 

WEIGH 

TS. 

Ih. 

oz. 

pwt. 

gr- 

A  Shekel, 

0 

9 

9 

2,0 

The  Maneh, 

2 

3 

6 

10,3 

A  Talent, 

113 

10 

1 

10,3 

MONB 

Y. 

dolls. 

cents. 

mills. 

A  Shekel, 

0 

50 

5 

The  Bekah,  {half  Sheh)       . 

0 

25 

3 

The  Zuza, 

0 

12 

5 

The  Gerah, 

0 

02 

5 

Maneh  or  Mina, 

25 

29 

6 

ATalent  of  Silver, 

1,157 

85 

7 

A  Shekel  of  Gold, 

8 

09 

4 

A  Talent  of  Gold, 

24,285 

71 

4 

Golden  Daric  or  Drachm, 

4 

85 

7 

dolls. 

cents. 

mills. 

Piece  of  Silver,  (Drachm) 

0 

14 

3 

Tribute  money,  (Didrachm) 

0 

28 

7 

Piece  of  Silver,  (Stater) 

0 

57 

4 

Pound,  (Mina) 

14 

35 

1 

Penny,  (Denarius) 

0 

14 

3 

Farthing,  [Assar'mm) 

0 

00 

6 

Farthing,  [Quadrands)     . 

0 

00 

3 

Mite,    '           ... 

0 

00 

1 

ARITH31ETIC. 

PART  FIRST. 


Arithmetic  is  the  science  of  numbers. 

A  unit  is  a  whole  thing  of  any  kind. 

A  fraction  is  a  part  of  a  thing. 

Thus  a  dollar  is  a  unit ;  a  man  is  a  unit ;  a  picture  is 
a  unit ;  a  bushel  of  apples  is  a  unit,  &c. 

A  half  of  an  apple  is  a  fraction  ;  a  quarter  of  a  dollar 
is  a  fraction  ;  a  third  of  a  loaf  of  bread  is  a  fraction,  6iC. 
Let  the  pupil  mention  other  units  and  fractions. 

If  an  apple  is  cut  into  two  equal  parts,  eacii  part  is 
called  one  half  of  the  apple.  If  it  is  cut  into  three  equal 
parts,  each  part  is  called  one  third.  If  it  is  divided  into 
four  equal  parts,  each  part  is  called  one  fourth.  If"  it  is 
divided  into^yp  equal  parts,  each  part  is  called  one  fifth,  iSfC. 

If  a  unit  is  divided  into  six  equal  parts,  what  is  one  of 
those  parts  called  ?  If  a  unit  is  divided  into  seven  equal 
parts,  what  is  one  of  those  parts  called  ?  If  a  unit  is  divi- 
ded into  eight  equal  parts,  what  is  one  of  those  parts 
called  ?  Into  nine  ?  Into  twenty  ?  Into  an  hundred  .'' 
Into  fifteen  ?     Into  twenty-two  ? 

How  many  halves  make  one  unit  1  How  many  thirds 
make  one  unit  ?  How  many  fourths  ?  How  many  fifths? 
How  many  sixths  ?  How  many  sevenths  ?  How  many 
eighths  ?  How  many  ninths  ?  How  many  tenths  ?  How 
many  twentieths  ?     How  many  hundredths  ? 

For  illustrating  the  exercises  which  immediately  follow, 
the  teacher  should  be  provided  with  a  proper  number  of 
the  several  coins  of  the  U.  S.  viz  :  eagles,  dollars,  dimes, 
cents  and  mills.  As  mills  have  never  been  coined,  round 
bits  of  stiff  paper  may  be  employed  to  represent  them. 
The  pupil  should  first  see  the  several  coins  and  learn  the 
value  of  them. 

Ten  mills  are  one  cent. 
Ten  cents  are  one  dime. 
Ten  dimes  are  one  dollar. 
Ten  dollars  are  one  eagle. 


22  ARITHMETIC.       FJRST  PART. 

How  many  mills  make  a  cent  ?  What  part  of  a  cent  is 
one  mill  ?  What  part  of  a  cent  is  two  mills  ?  What  part 
of  a  cent  is  three  mills?  What  part  of  a  cent  is  four 
mills  ?  What  part  of  a  cent  is  five  mills  ?  Six  mills  ? 
Seven  mills?     Eight  mills  ?     Nine  mills? 

How  many  cents  make  a  dime  ?  One  cent  is  what 
part  of  a  dime  ?  Two  cents  is  what  part  of  a  dime  ? 
Three  cents  is  what  part  of  a  dime  1  Four  cents  is  what 
part  of  a  dime?  Five  cents  ?  Six  cents  ?  Seven  cents? 
Eight  cents  ?     Nine  cents  ? 

How  many  dimes  make  a  dollar  ?  What  part  of  a  dol- 
lar is  one  dime  ?  What  part  of  a  dollar  is  two  dimes  ? 
What  part  of  a  dollar  is  three  dimes  ?  Four  dimes  ?  Five 
dimes?  Six  dimes?  Seven  dimes?  Eight  dimes? 
Nine  dimes  ? 

How  many  dollars  make  an  eagle  ?  One  dollar  is  what 
part  of  an  eagle  ?  Two  dollars  is  what  part  of  an  eS'gle  ? 
Three  dollars  ?  Four  dollars  ?  What  part  of  an  eagle  is 
five  dollars  ?  Six  dollars  ?  Seven  dollars  ?  Eight  dol- 
lars ?     Nine  dollars? 

The  same  thing,  may  be  considered  sometimes  as  a 
unit  and  sometimes  as  a  fraction — thus,  one  dollar  is  a 
unit  or  whole  thing  of  the  kind  or  order  called  dollars,  and 
one  dollar  is  also  the  te?ith  part  of  an  eagle,  or  the  fraction 
of  an  eagle.  One  cent  is  a  U7iit  or  whole  thing,  of  the  or- 
der of  cents,  and  one  cent  is  also  the  tenth  part  of  a  dime, 
or  ihe  fraction  of  a  dime.  One  mill  is  a  unit  of  the  order 
of  mills,  and  one  mill  is  the  tenth  part  of  a  cent,  or  the 
fraction  of  a  cent.  One  day  is  a  unit  or  whole  thing  of 
the  order  of  days,  and  one  day  is  also  the  seventh  fart  of 
a  week,  or  t\\e  fraction  of  a  week.  One  week  is  a  U7iit  or 
whole  thing  of  the  order  of  weeks,  and  one  week  is  the 
fourth  part  of  a  month,  or  the  fraction  o{  a.  monlh.  One 
month  is  a  unit  or  whole  thing  of  the  order  of  months,  and 
one  month  is  also  the  twelfth  part  of  a.  year,  or  the  fraction 
of  a  year. 

Of  what  order  is  one  dollar  a  unit  ?  Of  what  order  is  it 
a  fraction  ?  Of  what  order  is  one  cent  a  unit  ?  Of  what 
order  is  it  a  fraction  ?  Of  what  order  is  one  week  a  unit  ? 
Of  what  order  is  it  a  fraction  ?  Of  what  order  is  one  foot 
a  unit?  Of  what  order  is  it  a  Jraction  ?  One  day  is  a 
unit,  of  what  order,  and  a  fraction  of  what  order  ?  &c. 


ARITHMETIC.       FIRST  PART.  23 

What  is  half  of  four  cents  ?  What  is  a  third  of  six 
cents  ? 

Let  the  pupil  take  six  cents,  and  divide  them  into  three 
equal  portions,  and  then  tell  what  is  one  of  these  parts  ? 

What  is  -A  fourth  o(  eis;ht  cents?  Let  the  pupil  divide 
eight  cents  into  Jour  equal  portions,  and  tell  how  many  in 
each  portion. 

There  are  twice  six  cents  in  twelve  cents,  what  part  of 
twelve  is  six  cents  ? 

There  are  three  times  four  cents  in  twelve  cents,  what 
part  of  twelve  is  four  cents  ? 

There  are  three  times  five  cents  in  fifteen  cents,  what 
part  of  fifteen  is  five  cents  ? 

There  are  three  times  three  in  nine,  what  part  of  nine 
is  three  ? 

There  are  two  times  three  in  six,  what  part  of  six  is 
three  ? 

There  are  ybwr  times  two  in  eight,  what  part  of-  eight  is 
two? 

There  are  four  times  three  in  twelve,  what  part  of  twelve 
is  three  ? 

There  axe  five  times  six  in  thirty,  what  part  of  thirty  is 
six? 

There  are  three  times  seven  in  twenty-one,  what  part 
of  twenty-one  is  seven  ? 

There  are  four  times  six  in  twenty-four,  what  part  of 
twenty-four  is  six  ? 

There  are  six  times  seven  in  forty-two,  what  part  of 
forty -two  is  seven  ? 

What  part  of  twelve  is  three  ?     Is  four  ? 

What  part  of  nine  is  three  ? 

What  part  of  fifteen  is  three  ?     Is  five  1 

What  part  of  sixteen  is  four  ? 

What  part  of  eighteen  is  three  ?     Is  six  ? 

What  part  of  twenty-one  is  three  ?     Is  seven  ? 

What  part  of  twenty. four  is  six  ?     Is  four  ? 

What  part  of  twenty-eight  is  seven  ?     Is  four  ? 

What  part  of  thirty-two  is  eight  ?     Is  four  ? 

What  part  of  thirty-six  is  nine  ]     Is  four  ? 

If  an  apple  is  cut  into  two  equal  parts,  what  is  each  part 


24  ARITHMETIC.       FIRST  PART. 

called  ?  If  it  is  cut  into  three  equal  parts,  what  is  each 
part  called  ? 

The  more  parts  a  thing  is  divided  into,  the  smaller  these 
parts  must  be.  If  one  thing  is  divided  into  twice  as  many 
parts  as  another  thing,  each  part  is  twice  as  small. 

If  one  apple  is  cut  into  twice  as  many  pieces  as  another, 
how  much  smaller  is  each  piece  ?  How  much  larger  is  a 
half  than  a  fourth  ?  Ans.  There  are  twice  as  many  fourths 
as  halves  in  a  thing,  therefore  a  half  is  twice  as  large  as 
a  fourth; 

If  one  apple  is  cut  into  four  pieces,  and  another  into 
eight  pieces,  how  much  larger  are  the  fourths  than  the 
eighths  ?  Ans.  As  there  are  ttvice  as  many  pieces  when 
there  are  eighths,  as  when  there  are  fourths,  an  eighth  is 
twice  as  small  as  a  fourth. 

If  one  apple  is  cut  into  twelve  parts  and  another  into  six 
parts,  which  has  the  most  parts  and  which  has  the  largest 
parts  ?  How  much  larger  is  a  sixth  than  a  twelfth  ?  Ans. 
Twelve  is  twice  as  many  as  six,  therefore  a  sixth  is  twice  as 
large  as  a  twelfth. 

Which  is  the  largest,  a  fifth  or  a  tenth  1  How  much 
larger  is  a  fifth  than  a  tenth  ?       , 

Which  is  the  largest  a  seveiikth  or  a  fourteenth? 

How  much  smaller  is  a  fourteenth  than  a  seventh  ? 

Which  is  the  largest  a  third  or  a  fifth  ? 

Which  is  the  snsallest  a  half  or  a  fourth  ? 

Which  is  the  smallest  a  third  or  a  half?  Ans.  The 
more  pieces  there  are  the  smaller  they  must  be,  therefore 
a  third  must  be  smaller  than  a  half. 

If  one  apple  was  cut  into  four  pieces,  and  another  into 
six  pieces,  which  would  be  the  largest  a  fourth  or  a  sixth  ? 

Which  is  the  largest  a  sixth  or  a  ninth  ? 

Which  is  theiargest  a  fifth  or  a  fourth  ? 

Which  is  the  smallest  a  twelfth  or  a  tenth  ? 

Which  is  the  smallest  a  seventh  or  a  ninth  ? 

Which  is  the  smallest  an  eighth  or  a  seventh  ? 

Which  is  the  smallest  a  fifteenth  or  a  fifth  ? 

Which  is  the  largest  an  eighth  or  a  sixteenth? 

Which  is  the  largest  a  fifth  or  a  half? 

If  an  apple  is  divided  into  four  pieces,  what  is  each 
piece  ?     If  it  is  divided  into  twice  as  many  and  twice  as 


ADDITION.  25 

small  pieces,  how  many  are  there,  and  what  are  they 
called  '/ 

If  an  apple  is  divided  into  thirds,  what  would  you  change 
them  to,  to  make  them  twice  as  many  and  twice  as  small  / 

Make  two  fourths  twice  as  small  and  twice  as  many 
pieces  and  what  is  the  answer  ? 

What  part  of  a  thin<r  is  twice  as  small  as  a  half?  As 
a  third  1  As  a  fourth?  As  a  tifth?  As  a  sixth  ?  As  a 
«*eventh  ?  As  an  eighth  ?  As  a  ninth  1  As  a  tenth  ?  As 
in  eleventh  ?     As  a  twelfth  ?   ' . 

What  part  of  a  thing  is  twice  as  large  as  a  fourth?  As 
a  sixth  ?  As  an  eighth  ?  As  a  tenth  ?  As  a  twelfth  ? 
As  a  fourteenth  ?  As  a  sixteenth  ?  As  an  eighteenth  '! 
As  a  twentieth  ? 


ADDITION. 

Two  cents,  and  four  cents,  and  six  cents,  and  nine 
cents  are  how  many?*  Sixteen  cents,  and  twelve  cents, 
are  how  many  ? 

Five  dollars,  and  four  dollars,  andyiine  dollars,  are  how 
many  ?  /  .  \ 

Four  halves  of  ^n  apple,  and  six"  halves,  and  nine 
halves,  are  how  many  halves  ? 

Five  sixths  of  an 'apple,  and  four  sixths,  and  nine  sixths, 
are  how  many  sixths  ? 

Three  fifths  of  an  orange,  and  four  fifths,  and  nine  fifths, 
and  twelve  fifths,  are  how  many  fifths  ? 

Addition  is  uniting  several  numbers   in   one. 

When  ivhole  numbers  are  added,  it  is  Simple  Addition. 
When  fractions  are  added,  it  is  Fractional  Addition. 

Six  dimes,  five  dimes,  and  four  dimes  are  how  many  ? 

Seven  dollars,  eight  dollars,  and  nine  dollars  are  how 
many  ? 

Nine  cents,  three  cents,  twelve  cents,  and  ten  cents  are 
how  many  ? 

Four,  three,  and  seven  are  how  many  ? 

Eight,  five,  and  three  are  how  many  ? 

Nine,  six,  and  two  are  how  many  ? 


26  ARITHMETIC.       FIRST  PART. 

Seven,  five,  and  six  are  how  many  ? 

Eight,  nine,  and  two  are  how  many? 

Seven,  eight,  and  one  are  how  many? 

Eleven,  five,  and  six  are  how  many? 

Ten,  seven,  and  three  are  how  many? 

Ten  twentietlis,  six  twentieths,  and  five  twentieths  are- 
how  many  twentieths  ? 

One  thirteenth  of  a  unit,  four  thirteenths,  and  seven 
thirteenths  are  how  many  tliirteenths  ? 

One  fifth  of  a  dollar,  three  fifths,  and  eight  filths  are 
how  many  fifths  ? 

One  ninth  of  an  orange,  four  ninths,  and  six  ninths  are 
how  many  ninths  ? 

Seven  tenths  of  an  eagle,  two  tenths,  and  five  tenths 
are  how  many  tenths  ? 

Three  eighteenths,  nine  eighteenths,  and  four  eight- 
eenths are  how  many  eighteenths  ? 

Ten  thirtieths,  six  thirtieths,  and  five  thirtieths  are  how- 
many  thirtieths  ? 

Two  fourths,  six  fourths,  nine  fourths,  ten  fourths,  and 
five  fourths,  are  how  many  fourths  ? 

Sixteen  halves,  five  halves,  nine  halves,  and  six  halves, 
are  how  man}'  halves  ? 

Six  eighths,  four  eighths,  seven  eighths,  sixteen  eighths, 
are  how  many  eighths? 

The  ntnnber  made  by  adding  several  numbers  together, 
is  called  'he  sum. 

What  is  the  sum  of  four,  six,  nine  and  five  ? 

What  is  the  sum  of  four  tenths,  six  tenths,  and  nine 
tenths  ? 


SUBTRACTIOiN. 

If  you  take  two  cents  from  three  cents,  how  many  re- 
main ? 

If  you  take  three  dollars  from  six  dollars  how  many  re- 
main ? 

If  you  take  four  dollars  from  seven  dollars  how  many 
remain  ? 


SUBTRACTION.  27 

If  you  take  five  eagles  from  nine  eagles  how  nuiny  re- 
main ?    \ 

If  you  take  six  dimes  from  ten  dimes  how  many  re- 
main ? 

If  two  tenths  are  taken  from  four  tenths  how  many  re- 
n)ain  ? 

If  four  ninths  are  taken  from  eight  ninths  how  many  re- 
main ? 

If  two  tenths  are  taken  from  seven  tenths  how  many  re- 
main ? 

Subtraction  is  taking  one  number  from  another ,  to  find  the 
remainder. 

When  whole  numbers  are  subtracted  it  is  Sinjple  Sub- 
traction. When  fractions  are  sub.'racted,  it  is  Fractional 
Subtraction. 

What  is  the  remainder,  when  four  cents  are  taken  from 
nine  cents? 

What  is  the  remainder,  when  three  mills  are  taken 
from  eight  mills  ? 

What  is  the  remainder,  when  seven  dimes  are  taken 
from  twelve  dimes  ? 

What  is  the  remainder,  when  five  dollars  are  taken 
from  ten  dollars  ?  >, 

Five  from  eleven  1  Seven  from  thirteen  l  Eight  from 
twelve  ?  Five  from  fourteen  ?  Nine  from  sixteen  ?  Five 
from  twelve  ?     Eight  from  thirteen  ?     Ten  from  Twenty  ? 

W^hat  is  the  remainder,  when  two  sevenths  of  an  apple, 
are  taken  from  eight  sevenths  ?  When  four  sevenths  of 
a  dollar  are  taken  from  six  sevenths  ?  Eight  twelfths 
from  ten  twelfths?  Three  ninths  from  eight  ninths? 
Ten  twentieths  from  twelve  twentieths  ?  Six  elevenths 
from  ten  elevenths  ?  Seven  twelfths  from  twelve  twelfths  ? 
Eight  ninth's  from  thirteen  ninths  ?  Three  sevenths  from 
nine  sevenths  ?  Four  eighths  from  eleven  eighths  ?  Four 
thirds  from  twelve  thirds  1  Five  twentieths  from  seven 
twentieths  ? 

The  number  which  has  a  number  subtracted  from  it,  is 
called  the  minuend. 

The  number  which  is  to  be  subtracted  from  another  num- 
ber is  called  the  subtrahend. 

If  eight  is  subtracted  from  twelve,  what  is  the  subtra- 
hend and  what  is  the  minuend  ? 


28  ARITHMETIC.        FIRST    PART. 

If  four  tenths,  is  .subtracted  from  nine  tenths,  what  is 
the  subtra.hend  and  what  the  minuend? 

If  ten  cents  be  taken  from  thirteen  cents,  what  is  the 
subtrahend,  and  what  the  minuend  ? 


MULTIPLICATION. 

If  you  take  two  cents,  three  times,  what  is  the  amount 
of  the  whole? 

If  you  take  three  dollars,  ybt/T  times,  what  is  the  amount 
of  the  whole  ? 

If  you  take  half  of  an  apple  three  times,  what  is  the 
amount  ? 

If  you  take  two  thirds  of  a  AoWnx  four  times,  what  is  the 
amount  ? 

If  you  take  two  fourths  of  an  eagle,  six  times,  what  is  the 
amount  ? 

Multiplication  is  repeating  a  number  as  often  as  there- 
are  units  in  another  number. 

If  you  take  five  dollars ybwr  times,  what  is  the  amount  ? 

If  you  repeat  four  dollars  five  times,  what  is  the  amount  ? 

If  you  take  six  dollars  five  times,  what  is  the  amount  ? 

If  you  repeat  six  dollars  six  times,  what  is  the  amount  ? 
Seven  times  ?     Eight  times  ? 

If  you  take  seven  dollars  three  times,  what  is  the  amount  1 

If  you  repeat  seven,  four  times,  vfh^aX.  is  the  amount  ? 
Five  tmies  ?     Six  times  ?     Seven  times  ? 

If  you  repeat  eight  twice,  what  is  the  amount  ? 

If  you  repeat  eight  three  times,  what  is  the  amount  ? 
Four  times  ?      Five  times  ?      Six  times  ?      Seven  times  1 
Eight  times  ? 

If  you  repeat  nine  three  times,  what  is  the  amount  ?  &cc. 

If  you  take  one  fifth  of  a  dollar  six  times,  what  is  the 
amount  ?     Seven  times  ?     Eight  times  ?    Nine  times  ? 

If  you  repeat  two  sixths  of  a  dollar  three  times,  what  is 
the  amount  ? 


MULTIPLICATION. 


29 


If  you  repeat  two  sixths  of  a  thing  Jour  times,  what  is 
the  amount?  Five  times?  Six  times?  Seven  times? 
Eight  times  ? 

What  is  the  amount,  if  four  sevenths  be  repeated 
four  times  ?  Five  times  ?  Six  times  ?  Seven  times  ? 
Eight  times  ? 

What  is  the  amount  li  five  ninths  be  repeated  eight 
times  ?     Nine  times  ?     Ten  times  ?     Eleven  times  ? 

What  is  the  amount,  if  eight  twentieths  be  repeated 
seven  times  ?     Nine  times  ?     Eight  times  ?  &:c. 

The  number  to  be  repeated,  is  the  multiplicand. 

The  number  which  shows  how  often  the  muhiplicand  is  to 
bo  repeated,  is  called  the  mtdliplier. 

The  multiplier  and  midtiplicund  together,  are  called  the 
factors. 

The  answer  obtained  is  called  i\\Q  product. 

l[  eight  is  repeated  /our  times  what  is  the  |)roduct  ? 
What  is  the  multiplier  ?  The  multiplicand  ?  The  fac- 
tors  ? 

If  three  sixths  are  repealed  foiir  times  what  are  the  fac- 
tors ?     The  multiplier  ?     The  multiplicand  ? 

If  you  take  a.  fourth  of  twelve  and  repeat  it  three  times, 
what  is  the  multiplicand  ?  The  multiplier  ?  The  pro- 
duct ? 

If  you  take  a  sixth  of  eighleeli  and  repeat  it  three  times, 
what  is  the  product  ?  factors  ?  multiplier  ?  multipli- 
cand ? 

Simple  Multiplication  is  where  both  factors  are  whole 
numbers. 

Fractional  Multiplication  is  where  one  or  both  factors 
are  fractions. 

If  twelve  is  repeated  four  times,  is  it  simple  or  fraction- 
al multiplication  ? 

If  one  fourth  of  tvoelve  is  repeated  three  times,  is  it  sim- 
ple or  fractional  multiplication  ?  If  one  sixth  is  repeated 
seven  times,  which  kind  of  multiplication  is  it  ? 

Exercises  in  Simple  Multiplication. 

1.  If  a  man  spends  three  dollars  a  week,  how  much 
does  he  spend  a  month  ? 
3* 


30  ARITH3IETIC.       FIRST    PART. 

Let  the  pupil  state  the  sum  in  tliis  manner. 

As  there  are  four  weeks  in  a  month,  a  man  will  spend 
four  times  as  much  in  a  month,  as  in  a  week  ;  four  times 
three  is  twelve.     He  will  spend  twelve  dollars. 

Let  all  the  following  sums  be  stated  in  the  same  way. 
Both  teachers  and  pupils  will  find  great  advantage  in  be- 
ing particular  to  follow  this  method  of  stating. 

2.  If  a  man  spend  five  dollars  a  month,  how  much 
does  he  spend  in  a  j-ear  ? 

3.  If  a  man  can  make  eight  pens  in  a  minute,  how 
many  can  he  make  in  ten  minutes  ? 

4.  If  one  orange  cost  six  cents,  what  costs  eight  oran- 
ges ? 

5.  Eight  bays  have  seven  cents  apiece,  how  much  have 
all? 

6.  There  is  an  orchard  in  which  there  are  six  rows  of 
trees,  and  seven  in  each  row,  how  many  trees  in  the  or- 
chard ? 

7.  The  chess  board  has  eight  rows  of  blocks,  and  eight 
blocks  in  each  row,  how  many  blocks  in  the  whole  ? 

8.  Twelve  young  ladies  have  each  five  books  apiece, 
how  many  have  they  all  ? 

9.  l(  a  young  lady  spends  six  cents  a  week,  how  much 
does  she  spend  in  a  month  ? 

10.  There  are  nine  desks  in  a  school  room,  and  six 
scholars  at  each  of  the  desks,  how  many  are  in  the  room  ? 

11.  There  are  in  a  window  five  rows  of  panes  of  glass, 
and  seven  panes  in  each  row,  how  many  in  the  whole  ? 

12.  If  one  lemon  cost  four  cents,  how  much  will  twelve 
lemons  cost  ? 


EXERCISES    IN    FRACTIONAL    MULTIPLICATION. 

Multiplication  of  a  fraction  by  whole  numbers. 

1.  If  vou  repeat  one  half  four  times  what  is  the  pro- 
duct 1 

'Z.  If  you  multiply  three  fourths  by  seven,  what  is  the 
product  ? 

3.  What  is  two  thirds  multiplied  by  eight  ? 


DIVISION.  3 1 

4.  If  a  man  spend  two  twelfths  of  a  dollar  a  day,  how 
many  twelfths  does  he  spend  in  a  week  ? 

Ans.  As  there  are  seven  days  in  a  week,  a  man  spends 
seven  times  as  much  in  a  week  as  in  one  day.  Seven 
times  two  twelfths  is  fourteen  twelfths.  He  spends  four- 
teen twelfths  of  a  dollar  in  a  week. 

5.  If  a  man  gives  two  eighths  of  a  pound  of  meat  to  six 
persons,  how  many  eighths  does  he  give  away  ? 

6.  If  a  boy  gives  two  fourths  of  an  orange  to  seven  of 
his  companions,  how  many  fourths  does  he  give  away  ? 

7.  If  a  man  drinks  th-ee  Jourths  of  a  pint  of  brandy  a 
day,  how  many  fourths  does  he  drink  in  a  week  ? 

8.  What  is  three  times  tJu-ee  eighths  ?  Six  times  six 
sevenths ? 

9.  If  a  man  lays  by  two  eighths  of  a  dollar  a  day,  how 
much  does  he  save  in  a  week  ? 

10.  If  there  are  two  thirds  of  a  pound  of  meat  for  each 
one  in  a  family  of  seven,  'how  much  is  there  in  the  whole  ? 

11.  What  is  six  times  four  tenths  ? 

12.  What  is  nine  times  two  thirds  ? 
18.   What  is  seven  times  four  ninths? 

14.  What  is  eight  times  six  tenths  ? 

15.  What  is  twelve  times  two  fourths  7 

16.  What  is  nine  times  three  tenths? 

17.  What  is  five  times  three  sixteenths  ? 

18.  What  is  six  times  seven  twentieths  ? 

The  vmlUplication  of  whole  numbers  hy  fractions,  is  defer- 
red to  the  Second  Part,  because  it  involves  the  process  of 
Division,  which  must  first  be  explained. 


DIVISION. 

How  many  two  cents  are  there  in  four  cents  ? 
How  many  two  cents  in  six  cents  ? 
How  many  two  cents  in  eight  ? 
How  many  two  cents  in  ten  ? 
How  many  two  cents  in  twelve  ? 
How  many  three  cents  are  there  in  six  cents  ?    How 
many  in  nine  ?     How  many  in  twelve  ? 


32  ARITHMETIC.       FIRST   PART. 

How  many  four  cents  are  there  in  eight  ?  How  many 
in  twelve  ? 

How  many  five  cents  are  there  in  ten  ? 

What  part  part  of  two  cents  is  one  cent  ? 

What  part  of  four  cents  is  two  ?  What  part  of  six  is 
two  ?  What  part  of  eight  is  two  ?  What  part  of  ten  is 
two  ?     What  part  of  twelve  is  two  ? 

Three  cents  is  what  part  of  six  ?  Three  is  what  part  of 
nine  ?     Of  twelve  ? 

What  part  of  eight   is  four  ?     What  part  of   twelve 

is  four  ? 

What  part  of  five  cents  is  one  ?  What  part  of  five  is 
two  ?  What  part  of  five  is  three  ?  Four  ?  Five  ? 
Six?  &c. 

How  many  two  sixths  are  there  in  four  sixths  1 

How  many  three  fourths  are  there  in  six  fourths  ? 

How  many  four  twelfths  in  eight  twelfths  ? 

What  part  of  two  twelfths  is  one  twelfth  ? 

What  part  of  four  twelfths  is  two  twelfths  ? 

What  part  of  nine  twelfths  is  three  twelfths  ? 

Division  is  finding  how  of  fen  one  number  is  contained  in 
another,  and  thus  finding  what  part  of  one  number  is  another 
number. 

How  many  times  is  six  contained  in  twelve  ?  In 
eighteen  ? 

What  part  of  twelve  is  six  ?     What  part  of  eighteen  is 


six 


How  many  times  is  five  contained  in  ten  ?     In  fifteen  ? 

Five  is  what  part  often  ?     Of  fifteen  ? 

How  many  times  is  seven  contained  in  fourteen  ?  In 
twenty-one  ? 

What  part  of  fourteen  is  seven  ?  What  part  of  twen- 
ty-one is  seven  ? 

How  many  times  is  nine  contained  in  eighteen  ? 

How  many  times  is  ten  contained  in  twenty?  In  thir- 
ty  ?     In  forty  ? 

What  part  of  sixteen  is  four  ? 

What  part  of  eighteen  is  six  ? 

What  part  of  sixteen  is  eight  ? 

One  is  what  part  of  thirty  ?    Two  is  what  part  of  thir- 


DIVISION.  33 

ty  ?  Three  is  what  part  of  tliirty  ?   Six  ?  Eight  ?  Eleven  ? 
Fourteen  ?     Twenty  is  what  part  of  thirty  ?  &;c. 

How  many  two  sevenths  are  there  in  ten  sevenths  ? 

How  many  three  eighths  are  there  in  nine  eighths  ? 

How  many  six  tenths  in  eighteen  tenths  ? 

How  many  seven  ninths  in  twenty-one  ninths  1 

How  many  five  elevenths  in  twenty  elevenths  ? 

How  many  three  eighteenths  arc  there  in  twelve  eight- 
eenths  1 

Two  sixths  is  what  part  of  four  sixths  ? 

Two  sevenths  is  what  part  often  sevenths? 

Three  eighths  is  what  part  of  nine  eighths  ? 

What  part  of  eighteen  tenths  is  six  tenths  ? 

What  part  of  fourteen  ninths  is  seven  ninths  ? 

What  part  of  fifteen  elevenths  is  five  elevenths? 

What  part  of  twelve  eighteenths  is  three  eighteenths  ? 

The  number  which  is  divided  is  called  the  Dividend. 

The  number  by  which  you  divide  is  called  the  Divisor. 

The  answer  is  called  the  Quotient. 

If  you  find  how  many  times  three  there  are  in  twelve, 
which  is  the  Divisor  1     The  Dividend  1     The  Quotient  ? 

If  twelve  is  divided  by  six,  which  is  the  Dividend  ?  The 
Divisor  ?     The  Quotient  ? 

When  whole  numbers  are  divided  by  whole  numbers,  it 
is  called  Sim-ple  Division. 

When  either  the  divisor  or  dividend  is  a  fraction,  it  is 
called  Fractional  Division. 


Exercises  in  Simple  Division. 

1.  If  5'ou  divide  12  cents  equally  among  three  boys,  how 
many  will  each  one  have  ? 

Ans.  Each  one  will  have  as  many  as  there  are  threes 
in  twelve  ;  or  four  cents. 

2.  If  there  are  forty-eight  panes  of  glass  in  a  window, 
and  there  are  eight  panes  in  each  row,  how  many  rows 
are  there  ? 

Ans.  As  many  as  there  are  eights  in  forty-eight ;  ov 
six  rows. 

3.  How  much  broadcloth,  at  six  dollars  a  yard,  can 
you  buy  for  twenty-four  dollars  1 


34  ARITHMETIC.       FIRST   PART. 

4.  How  many  hours  would  it  take  you  to  travel  twen- 
ty-one miles,  if  you  travelled  three  miles  an  hour  1 

5.  If  you  divided  thirty-six  apples  equally  among  four 
boys,  how  many  would  you  give  them  apiece  ? 

6.  How  many  pounds  of  raisins,  at  nine  cents  a  pound, 
can  you  buy  for  sixty-three  cents  ? 

7.  How  many  reams  of  paper,  at  seven  dollars  a  ream, 
can  you  buy  for  forty-nine  dollars  ? 

8.  A  man  agreed  to  work  eight  months,  for  seventy -two 
dollars,  how  much  did  he  receive  a  month  ? 

9.  If  you  buy  a  bushel  of  pears  for  forty-eight  cents, 
how  much  is  it  a-  peck  ? 

10.  If  there  are  six  shillings  in  a  dollar,  how  many  dol- 
lars in  thirty-six  shiUings  ? 

11.  Four  men  bought  ahorse  for  forty-eight  dollars, 
what  did  each  man  pay  1 

12.  A  man  gave  sixty-three  cents  for  a  horse  to  ride 
nine  miles,  how  much  was  that  for  each  mile  ? 

13.  A  man  agreed  to  pay  eight  cents  a  mile  for  a  horse, 
and  he  paid  sixty-four  cents,  how  many  miles  did  he  go  ? 

14.  A  man  had  forty-two  dollars,  which  he  paid  for 
wood,  at  seven  dollars  a  cord,  how  many  cords  did  he 
buy? 

15.  Two  boys  are  running,  and  are  forty-eight  rods 
apart.  The  hindermost  boy  gains  upon  the  other,  three 
rods  a  minute,  in  how  many  minutes  will  he  overtake  the 
foremost  boy  ? 

16.  A  vessel  contains  sixty-three  gallons,  and  dischar- 
ges seven  gallons  an  hour,  in  how  many  hours  will  it  be 
emptied  ? 

17.  If  you  wish  to  put  sixty-four  pounds  of  butter  in 
eight  boxes,  how  many  pounds  would  you  put  in  each 
box? 


EXERCISES    IN    FRACTIONAL    DIVISION. 

Division  of  whole  numbers  by  Fractions. 

1.  How  many  halves  are  there  in  six  oranges? 

2.  How  many  thirds  are  there  in  four  apples  ? 


DIVISION.  35 

Ans.  One  apple  has  three  thirds, ybwr  apples  haxe  four 
times  as  many,  or  twelve  thirds. 

3.  How  many  fourths  are  there  in  three  oranges? 

4.  How  many  fifths  are  there  in  four  apples  1 

5.  How  many  sixths  are  there  in  two  oranges  ? 

G.  How  many  half  dollars  are  there  in  four  dollars? 

7.  How  many  quarters  of  a  dollar  in  five  dollars? 

8.  How  many  half  eagles  in  eight  eagles  ? 

9.  In  two  dollars  how  many  thirds  of  a  dollar  ? 

10.  If  there  are  six  one  thirds  in  two  dollars,  how  many 
two  thirds  are  there  ? 

Ans.  Tfiere  are  only  half  as  many  two  thirds  as  there 
are  cwie  thirds,  or  three  two  thirds. 

11.  In  two  dollars,  how  many  one  sixths?  How  many 
two  sixths  ? 

12.  A  man  divided  two  dollars  among  his  workmen, 
and  gave  them  a  third  of  a  dollar  apiece,  how  many  work- 
men had  he  ? 

13.  A  man  divided  four  dollars  equally  among  his  chil- 
dren, and  gave  them  each  tico  thirds  of  a  dollar,  how  ma- 
ny children  had  ho  ? 

Ans,  As  many  children  as  there  are  t^co  thirds  in  four 
dollars.  In  four  dollars  there  are  twelve  one  thirds. 
There  are  half  as  many  two  thirds,  or  six.  He  had  six 
children. 

14.  If  a  man  gave  two  sevenths  of  a  dollar  to  each  of 
his  servants,  and  gave  away  in  the  whole  four  dollars, 
how  many  servants  had  he  ? 

15.  How  many  two  sixths  in  four  ? 

16.  How  many  two  eighths  in  four  ? 

17.  How  many  two  thirds  in  eight  ? 

18.  How  many  two  ninths  in  six? 

19.  How  many  two  twelfths  in  two  ? 

20.  How  many  two  twelfths  in  four  ? 

Dii-ision  of  Fractions  by  whole  numbers. 

In  dividing  fractions  by  whole  numbers,  we  do  not  find 
h(yw  many  times  a  whole  thing  is  contained  in  a  part  of  the 
same  thing,  for  that  would  be  absurd  ;  but  we  find  what 
part  of  once,  a  whole  number  is  contained  in  a  fraction. 

Thus  if  we  wish  to  divide  one  halj  by  one,  we  say,  one 
unit  is  contained  in  one  half,  not  once,  but  one  half  of  once. 


36  ARITHMETIC.       FIRST  PART. 

1.  One  is  contained  in  one  fourth,  what  part  of  once  ? 

2.  One  is  contained  in  one  fifth,  what  part  of  once  ? 

3.  One  is  contained  in  one  sixth,  what  part  of  once  1 

4.  One  is  contained  in  one  seventh,  what  part  of  once  ? 

5.  One  is  contained  in  one  eighth,  what  part  of  once  ? 

6.  One  is  contained  in  one  ninth,  what  part  of  once  ? 

7.  One  is  contained  in  one  tenth,  what  part  of  once  ? 

8.  One  is  contained  in  one  eleventh,  what  part  of  once  7 

9.  One  is  contained  in  one  twelfth,  what  part  of  once  1 

10.  If  you  divide  one  fourth,  hy  one,  which  is  the  divi- 
sor ?     The  dividend  ?     What  is  the  quotient  ? 

11.  If  vou  divide  one  sixth  hy  one,  what  is  the  quotient  ? 
The  divisor  ?     The  dividend  ? 

12.  If  you  divide  one  third  by  one,  what  is  the  quo- 
tient  ?     The  divisor  ?     The  dividend  ? 

13.  If  one  fourth  contains  one,  a  fourth  of  once,  what 
part  of  once  does  two  fourths  contain  it? 

Ans.  Tv.'ice  as  much,  or  two  fourths  of  once. 

14.  \Hwo  sixths  is  divided  by  one,  wiiat  is  the  answer  ? 
Ans.     Tico  sixths  of  once. 

15.  Ttco  eighths  contain  one,  what  part  of  once  ?  Six 
eighths  contain  one,  what  part  of  once  ? 

16.  Two  txi-elfths  contain  one,  what  part  of  once  ? 
Four  twelfths  contain  one,  what  part  of  once  ? 

17.  Eight  twelfths  contains  one,  what  part  of  once  ? 

18.  Six  twelfths  contains.one,  what  part  of  once  ? 

If  six  twelfths  contains  one,  six  twelfths  of  once,  it  would 
contain  two,  on\y  half  a.s  often,  or  three  twelfths  of  once. 

19.  Four  eighths  contains  one,  what  part  of  once  ? 
Contains  two,  wliat  part  of  once?  It  contains  two,  only 
half  as  often,  or  two  eighths  of  once. 

20.  Six  tenths  contains  one,  what  part  of  once  1  Con- 
tains  Zk'o,  what  part  of  once  ? 

21.  Eight  tenths  contains  one,  what  part  of  once  ? 
Contains  ttco,  what  part  of  once  ? 

2*^.  Four  eighths  contains  one,  what  part  of  once  ? — 
Contains  two,  what  part  of  once? 

23.  Six  elevenths  contains  one,  what  part  of  once  1 — 
Contains  two,  what  part  of  once  ? 

24.  Eight  twelfths  contains  one,  what  part  of  once  ? 
Contains  two,  what  part  of  once  ? 


REDUCTION. 


37 


REDUCTION. 

One  dime  is  how  many  cents  ?     How  many  mills  ? 

One  unit  of  the  order  of  dollars,  is  how  many  units  of 
the  order  of  dimes?  How  many  of  the  order  of  cents? 
How  many  of  the  order  of  mills  ? 

One  eagle  is  iiow  many  dollars  ?  How  many  dimes  ? 
Cents  ? 

One  unit  of  the  order  of  dimes  is  how  many  units  of  the 
order  of  cents? 
Reduction  is  changing  units  of  one  order,  to  those  of  another. 

A  unit  of  the  order  of  eagles  is  how  many  units  of  the 
order  of  dollars  ?     Of  dimes  1 

Two  eagles  are  how  many  dollars  ?     How  many  dimes  ? 

How  many  dollars  in  two  hundred  cents  ? 

How  many  dollars  in  twenty  dimes  ? 

Thirty  units  of  the  order  of  dimes,  is  how  many  units 
of  the  order  of  dollars  ? 

Two  pints  are  one  quart. 
Eight  quarts  are  one  peck. 
Four  pecks  are  one  bushel. 

Two  units  of  the  order  of  quarts,  are  how  many  units 
of  the  order  of  pints  ? 

Eight  pints  are  how  many  quarts  ? 
Two  bushels  how  many  pecks  ? 
Eight  pecks  how  many  bushels  ? 

Three  barley-corns  are  one  inch. 
Twelve  inches  are  one  foot. 
Three  feet  are  one  yard. 

One  inch  how  many  barley-corns  ?  Two  inches  are 
how  many  ? 

Twelve  barley-corns  how  many  inches  ? 

One  foot  how  many  inches?     Three  feet  how  many? 

One  yard  is  how  many  feet  ?  How  many  inches  ? 
How  many  barley-corns  ? 

Two  yards  are  how  many  feet  ?  How  many  inches  ? 
How  many  barley-corns  ? 

Three  yards  are  how  many  feet  ?  How  many  inches  ? 
How  many  barlev-corns  ? 

4 


38  ARITHMETIC.       FIRST  PART. 

How  many  feet  are  there  in  five  yards  ?     How  many 
inches  in  five  yards  ?     How  many  barley-corns  ? 

How  many  barley-corns  are  there  in  seven  yards  ? 

From  the  preceding  exercises,  you  learn  that  a  unit  of 
one  order  may  contain  several  units  of  another  order. 

What  do  you  learn  from  the  preceding  exercises  1 

How  many  units  of  the  order  of  cents,  are  there  in  one 
unit  of  the  order  of  dimes  ? 

How  many  units  of  the  order  of  dollars,  are  there  in  one 
unit  of  the  order  of  eagles  1 

How  many  units  of  the  order  of  mills,  are  there  in  one 
unit  of  the  order  of  cents  ? 

How  many  units  of  the  order  of  pints,  are  there  in  one 
unit  of  the  order  of  quarts  ? 

How  many  units  of  the  order  of  pecks,  are  there  in  one 
unit  of  the  order  of  bushels  ? 

How  many  units  of  the  order  of  barley-corn.s,  are  there 
iu  one  unit  of  the  order  of  inches  ? 

How  many  units  of  the  order  of  feet,  are  there  in  one 
unit  of  the  order  of  yards  1 

How  many  units  of  the  order  of  da)^s,  are  there  in  one 
unit  of  the  order  of  weeks  ? 

How  many  units  of  the  order  of  weeks,  in  one  unit  of 
the  order  of  months  ? 

Change  two  units  of  the  order  of  dimes,  to  units  of  the 
order  of  cents. 

Change  twenty  units  of  the  order  of  cents,  to  units  of 
the  order  of  dimes. 

Change  three  units  of  the  order  of  yards,    to  units  of 
the  order  offset. 

Change  nine  units  of  the  order  of  feet,  to  units  of  the 
order  of  yards. 

Change  ten  units  of  the  order  of  pints,  to  units  of  the  or- 
der of  quarts. 

Change  five  units  of  the  order  of  quarts,  to  units  of  the 
order  of  pints. 

Change  twenty-one  units  of  the  order  of  days,  to  units 
of  the  order  of  weeks,  &;c. 

When  units  of  one  order  are  changed  to  units  of  a  high- 


REDUCTION. 


39 


er  order,  the  process  is  called  Reduction  ascending ;  and 
when  units  of  one  order  are  changed  to  those  of  a  lower 
order,  the  process  is  called  Reduction  descending. 

If  twenty  cents  are  changed  to  dimes,  which  kind  of 
reduction  is  used  ? 

If  twenty  cents  are  changed  to  mills,  which  kind  of  re- 
duction is  used  ? 

If  four  gallons  are  changed  to  pints,  which  reduction  is 
used  ? 

If  eight  feet  are  changed  to  inches,  which  kind  of  re- 
duction is  used  ? 

In  changing  twelve  barley-corns  to  inches,  which  kind 
of  reduction  is  used  1 

In  changing  fourteen  days  to  weeks,  which  reduction  is 
used  ? 

In  changing  five  hours  to  minutes,   which  reduction 
is  used  ? 

In  changing  one  hundred  and  twenty  minutes  to  hours, 
which  reduction  is  used? 

Reduce  three  dimes  to  cents  ;  to  mills.  Which  kind 
of  reduction  is  it  ? 

Reduce  three  hundred  mills  to  cents ;  to  dimes  ;  and 
which  kind  of  reduction  is  it  ? 

Reduce  ihree  hundred  mills  to  dollars,  and  which  kind 
of  reduction  is  it  ? 

Reduce  two  halves  to  quarters,  and  which  kind  of  re- 
duction is  it  ? 

Ans.  As  a  half  is  of  more  value,  it  is  a  higher  order  than 
a  quarter,  therefore  it  is  reduction  ascendmg. 

In  performing  this  last  exercise,  the  pupil  will  find  the 
necessity  for  the  following  distinction  in  regard  to  units. 

A  unit  has  been  defined  as  "  any  wliole  thing  of  a 
kind,"  and  a  fraction  is  defined  as  "  a  part  of  a  thing." 

But  it  is  very  often  the  case,  that  fractions  are  consid. 
ered  as  units.  Thus  when  we  reduce  quarters  to  halves, 
and  halves  to  quarters,  we  cliange  units  of  the  order  called 
quarter,  to  units  of  the  order  called  half. 

When  we  say  a  whole  quarter  of  an  apple,  and  a  half  a 
quarter  of  an  apple,  we  think  of  a  quarter  as  a  whole  thing 
of  iis  kind. 

The  difference  between  the  two  kinds  of  units  is  this: 


40  ARITHMETIC.       FIRST  PART. 

When  we  think  of  a  whole  quarter,  we  think  of  another 
thing  of  which  the  quarter  is  a  part.  We  think  of  it  as  a 
whole  thing  in  one  respect,  and  as  a  part  of  a  thing  in 
another  respect.  But  when  we  think  of  a  whole  apple,  we 
do  not  necessarily  think  of  another  thing  of  which  it  is  a 
part. 

When  we  think  of  a  /taZf  of  a  loaf  of  bread,  do  we  think 
of  something  of  which  the  half  is  a  part? 

When,  we  think  of  a  biscuit,  do  we  necessarily  think  of 
something  of  which  it  is  a  part  ? 

When  we  think  of  a  third  of  an  orange,  do  we  necessa- 
rily think  of  something  of  which  it  is  a  part  ? 

When  we  think  of  a  house,  do  we  necessarily  think  of 
any  thing  of  which  it  is  a  part  ? 

Those  units  which  do  not  require  us  to  think  of  any  oth- 
er thing  of  which  they  are  parts,  are  called  whole  numbers, 
and  those  units  which  do  require  us  to  think  of  other 
things  of  which  they  are  parts,  are  CdWeA  fractions. 

What  is  the  difference  between  units  that  are  whole 
numbers,  and  units  that  are  fractions  1 

Reduce  two  yards  to  quarters,  and  which  kind  of  re- 
duction is  it  ? 

Reduce  twenty-four  inches  to  feet,  and  which  kind  of 
reduction  is  it  ? 

Reduce  three  feet  to  inches,  and  which  kind  of  reduc- 
tion is  it  ? 

Which  is  of  highest  value,  a  half  or  a  quarter  ? 

Reduce  eight  quarters  to  halves,  and  which  kind  of  re- 
duction is  it  ? 

Reduce  two  halves  to  quarters,  and  which  kind  of  re- 
duction  is  it  ? 

Reduce  sixteen  quarters  to  halves,  and  which  kind  of 
reduction  is  it  1 

Reduce  two  fifths  to  tenths  ;  six  tenths  to  fifths  ;  eight 
tenths  to  fifths  ;  twelve  tenths  to  fifths  ;  three  fifths  to 
tenths  ;  six  fifths  to  tenths. 

Reduce  one  seventh  to  fourteenths  ;  four  fourteenths 
to  sevenths ;  four  sevenths  to  fourteenths  ;  eight  four, 
teenths  to  sevenths. 


SUMMARY  OF  DEFINITIONS.  41 

Reduce  two  sixths  to  twelfths  ;  four  twelfths  to  sixths  ; 
eight  twelfths  to  sixths ;  five  sixths  to  twelfths  ;  four 
twelfths  to  sixths. 


SUMMARY  OF  DEFINITIONS. 

A  unit  is  any  whole  thing  of  a  kind. 

X  fraction  is  a  part  of  a  thing. 

Addition  is  uniting  several  numbers  in  one. 

Subtraction  is  taking  one  number  from  another,  to  find 
the  remainder. 

The  largest  number  is  the  minuend,  the  smallest  num- 
ber is  the  subtrahend, 

MuUipJication  is  repeating  one  number  as  often  as 
there  are  units  in  another  number. 

The  mtdlipUcand  is  the  number  to  be  repeated ;  the 
multiplier  is  the  number  which  shows  how  often  the  multi- 
plicand is  to  be  repeated  ;  the  factors  are  both  the  multi- 
plier and  miiltiphcand  ;  and  the  product  is  the  number  ob- 
tained by  multiplying. 

Division  is  finding  how  often  one  number  is  contained 
in  another  number,  and  thus  finding  what  part  of  one  num- 
ber, is  another  number. 

The  dividend  is  the  number  to  be  divided.  The  divi- 
sor is  the  number  by  which  you  divide.  The  quotient  is 
the  answer  obtained  by  dividing. 

Reduction  is  changing  units  of  one  order,  to  units  of 
another  order. 

Reduction  ascending,  is  changing  units  of  a  lower,  to  a 
higher  order. 

Reduction  descending  is  changing  units  of  a  higher,  to 
a  lower  order. 

Note  to  Teachers. — A  review  of  this  First  Part,  will 
be  found  more  useful  than  an  increased  number  of  ex- 
amples. 

4* 


ARITHMETIC. 

SECOND  PART. 


NUMERATION. 

Numeration  is  the  art  of  expressing  numbers  by  wordsy 
or  by  figures. 

Figures  are  sometimes  called  numbers,  because  they 
are  used  to  represent  numbers.  Thus  the  figure  4,  is  oft- 
en called  the  number  four,  because  it  is  used  to  represent 
that  number. 

There  are  thirty-five  words,  that  are  commonly  used  in 
numeration ;  viz  :  one,  two,  three,  four,  five,  six,  seven, 
eight,  nine,  ten,  eleven,  twelve,  thirteen,  fourteen,  fifteen,  six- 
teen, seventeen,  eighteen,  nineteen,  twenty,  thirty,  forty, fif- 
ty, sixty,  seventy,  eighty,  ninety,  hundred,  thousand,  million, 
hUlion,  trillion,  quadrillion,  quintiUion,  sextillion. 

Those  words  ending  in  ietn,  are  the  words  two,  three, 
four,  &c.  with  teen,  which  signifies  and  ten,  added  to 
them. 

What  is  the  meaning  o^ fourteen  ?  Ans.  Four  and  ten. 
What  is  the  meaning  o'i  thirteen?  of  nineteen?  oi seven- 
teen  ? 

Those  ending  in  ty,  are  the  words  two,  three,  four,  «fec. 
with  ty,  which  means  tens,  added  to  them. 

What  is  the  meaning  of  sixty  ?  of  seventy  ?  of  eighty  ? 
of  twenty?  ofthiity? 

The  words  of  spoken  numeration  would  be  more  uni- 
form, \i eleven  and  toefoe,  had  been  called  oneteennnAtioo- 
teen. 

The  Latin  and  Greek  numerals  are  so  often  used  in  the 
various  sciences,  that  it  is  importnnt  for  pupils  to  learn 
their  names.  They  are  therefore  put  down  with  the  fig- 
ures,  and  the  English  names.  The  figures  are  called 
Arabic,  because  first  introduced  into  Europe  from  Ara- 
bia. 


NUMERATION. 


43 


EN6LISII,    LATIN,    AND    GREEK    NUMERALS. 

Arabic  Figures.    English  Names.  Latin  Names.  Greek  Names. 

•^  '^  Eis. 

Duo. 
Treis. 
Tessares. 
Pente. 
Hex. 
Hcpta. 
Okio. 
Ennea. 
Deka. 
Eiideka. 
Dodcka. 
Dckalieia, 
Dekatessares. 
Dekapente. 
Dekaex. 
Dekaepta. 
Dekaocto. 
Dekaennea. 
Eikopi. 
TriaJ<onta. 
Tcsserakonta. 
Pentakonta. 
Hexakonta. 
HebdoTiiekonta. 
Ogdoekonta. 
Eunenekonta. 
Hekaton. 
Chilio. 

Billion,  Trillion,  Q,uddrillion,  Q,uintillion,  Sextillion,  &c.  are 
made  by  adding  cipheis  to  1. 

If  any  higher  number  than  sextillion  is  to  he  expressed, 
the  names  are  made  by  the  Latin  numerals,  with  illion 
added  to  them  ;   as  sepliUiun,  ocliUion,  &c. 

A  unit  has  been  defined  as  "  a  single  thing  of  any 
kind." 

But  a  unit  of  one  kind,  maybe  made  up  of  several  units 
of  another  kind.  Thus  the  unit  one  dollar  is  made  up  of 
ten  units,  of  the  kind,  or  order  called  diines ;  and  one 
dime  is  made  up  often  units  of  the  order  called  cents. 

A  unit  which  is  of  the  most  value,  is  called  a  unit  of  a 
hinher  order. 


1 

One. 

Unus. 

2 

Two. 

Duo. 

3 

Three. 

Tres. 

4 

Four. 

Q,uatuor. 

5 

Five. 

Q,uinque. 

6 

Six. 

Sex. 

7 

Seven. 

Septem. 

8 

Eight. 

Octo. 

9 

Nine. 

Novem. 

10 

Ten. 

Decern. 

11 

Eleven. 

Undecim. 

12 

Twelve. 

Duodeciin. 

13 

Thirteen. 

Tredecim. 

14 

Fourteen. 

duatuordecim. 

15 

Fifteen. 

Q,uindecim. 

16 

Sixteen. 

Sexdecim. 

17 

Seventeen. 

Septendecim. 

18 

Eighteen. 

Octodecim. 

19 

Nineteen. 

Novemdecim, 

20 

Twenty. 

Viginti. 

30 

Thirty. 

Triginta. 

40 

Furtv'. 

Q,uadraginta. 

50 

Fifty. 

Q,uinquaginti. 

60 

Sixty. 

Sexaginta. 

70 

Seventy. 

Septuaginta. 

80 

Eighty. 

Octoginta. 

90 

Ninety. 

Nonaginta. 

100 

Hundred. 

Centum. 

1000 

Thousand. 

Mille. 

100()0(X) 

Million. 

44  ARITH3IETIC.       SECOND  PART 

Which  unit  is  of  the  highest  order,  a  dollar  or  a  cent? 

How  many  units  of  the  order  of  dimes,  are  there  in  one 
unit  of  the  order  of  dollars  ? 

How  many  units  of  the  order  oi  mills,  make  one  unit  of 
the  order  oi cents? 

How  many  units  of  the  order  o^  cents,  make  one  unit  of 
the  order  of  dimes  ? 

Every  Jigure  represents  a  certain  rmmJer ;  but  the  num- 
ber it  represents,  depends  upon  the  0)-der  in  which  it  is 
placed. 

If  the  figure  (2)  stands  alone,  it  represents  two  umts, 
and  is  said  to  be  in  thejirstov  unit  order. 

But  if  it  has  a  figure  to  the  right  of  it,  thus  (20)  it  rep- 
resents two  tens,  or  twenty,  and  is  in  the  second  order,  or  the 
order  of  tens. 

The  cipher  is  put  to  the  right,  to  make  the  2  stand  in 
the  order  of  tons,  and  to  show  that  there  are  no  units  of 
the  unit  order.  If  some  figure  was  not  placed  there,  the  2 
would  be  in  the  tinii  order. 

If  the  figure  2  has  two  figures  to  the  right  of  it,  thus 
(200)  it  represents  two  hundreds,  and  stands  in  the  third 
07-der,  oi  i\\e  order  of  hundreds. 

From  this  it  appears,  that  in  numeration,  the  number  ex. 
pressed  by  any  figure,  depends  upon  the  order  in  ichich  it 
stands. 

The  number  which  any  figure  expresses  when  it  is  con- 
sidered alone,  is  called  its  simple  value.  The  number  it 
expresses  when  placed  with  other  figures,  is  called  its  lo- 
cal value. 

When  2  is  considered  alone,  what  is  its  simple  value  ? 
When  it  is  considered  as  in  the  order  of  tens,  what  is  its 
local  value  ?  When  ia  the  order  of  iiundreds,  what  is  its 
local  value  ? 

Questions. — What  does  every  figure  rej)resent  ?  W^hat 
does  the  number  which  any  figure  represents  depend  up- 
on ?  If  a  figure  stands  alone,  in  what  order  is  it  ?  If  it 
has  one  figure  at  the  right  of  it,  in  what  order  is  it?  If  it  has 
two  figures  cit  the  right  of  it,  in  what  order  is  it?  In  this 
number,  (234)  m  what  order  is  the  2  ?  the  3  ?  the  4  T 
Write  one  ten. — Why  is  the  cipher  used  ?  What  would 
the  number  be,  if  the  cipher  were  removed  ? 


NUMERATION.  45 

Write  one  ten  and  one  unit.  What  is  the  name  of  this 
number  ?     Ans.  Eleven. 

Write  one  ten  and  two  units.  What  is  the  name  of  this 
number? 

Write  one  ten  and  three  units.     What  is  the  name  ? 

Write  one  ten  and  four  units.     What  is  the  name  ? 

Write  one  ten  and  five  units.     What  is  the  name  ? 

Write  one  ten  and  six  units.     What  is  the  name  ? 

Write  one  ten  and  seven  units.     W^hat  is  the  name  ? 

Write  one  ten  and  eigjit  units.     Wiiat  is  the  name  7 

Write  one  ten  and  nine  units.     What  is  the  name  ? 

Write  two  tens.     What  is  the  name  ?     Ans.   Twenty. 

Write  three  tens.     What  is  the  name  ? 

'W  rite  four  tens ;  five  tens ;  six  tens;  seven  tens;  eight 
tens  ;  nine  tens ;  and  tell  their  names. 

Write  one  of  the  order  oihundreds. 

Write  two  of  the  order  of  hundreds ;  one  of  the  order  of 
ler,s ;  and  four  of  the  order  ofunifs. 

Write  two  of  the  order  of  hundreds ;  no  tens  ;  four 
units. 

Write  4  hundreds,  no  tens,  no  units. 

Write  two  hundreds,  eight  tens,  and  nine  units.  Sev- 
en hundreds,  six  tens,  and  three  units.  Two  tens,  and 
two  units.  Nine  tens,  and  six  units.  Four  hundreds, 
six  tens,  and  four  units.  Five  hundreds,  five  tens,  and 
five  units.  Nine  hundreds,  seven  tens,  and  three  units. 
Four  hundreds,  eight  tens,  and  four  units.  Eight  hun- 
dreds, nine  tens,  and  nine  units.  Two  hundreds,  six  tens, 
and  three  units.  One  hundred,  two  tens,  and  three 
units.  Two  hundreds,  five  tens,  and  seven  units.  One 
ten,  and  three  units.  Seven  tens,  and  three  units.  Nine 
hundreds,  nine  tens,  and  nine  units. 

In  reading  numbers,  we  can  either  mention  each  order 
separately,  or  simply  mention  the  name.?  of  the  numbers. 

Thus  we  can  call  this  number,  (21)  either  two  tens,  and 
one  unit,  or  twenty-one. 

This  number  (305)  can  be  read,  3  hundreds  ;  0  tens  ; 
5  units  ;  or  it  can  be  called  three  hundred  and  five. 

The  following  numbers  are  read  both  ways,  thus  ; 

10  One  ten  ;  no  units  ;  or  ten. 

1 1  One  ten ;  one  unit ;  or  eleven. 


46  ARITHMETIC.       SECOND    PART. 

208  Two  hundreds  ;  no  tens  ;  eight  units  ;  or  tvM  hun- 
dred and  eight. 

40  Four  tens  ;  no  units  ;  or  forty. 

Let  the  pupil  read  the  following  numbers  both  ways. 

111.  203.  41.  37.  542.  1.  11.  12.  60. 
300.     101.     639.     700.     305. 

In  this  number,  (203)  why  is  the  cipher  put  in?  What 
would  the  number  be  if  it  were  left  out? 

In  numeration,  every  unit  of  one  order,  is  considered 
as  composed  often  units  of  a  lower  order;  just  as  in  the 
coins  of  this  country,  ten  units  of  the  order  of  ccHf*,  make 
one  unit  of  the  order  of  dimes,  and  ten  units  of  the  order 
of  dimes,  make  one  unitof  the  order  of  dollars. 

So  in  numeration,  ten  units  of  the  order  of  units,  make 
07ie  ten  ;  ten  units  of  the  order  of  tens,  make  one  unit  of 
the  order  of  hundreds ;  ten  hundreds,  make  one  unit  of 
the  order  of  thousands ;  ten  thousands  make  one  of  the  ov- 
dev  of  tens  of  thousands ;  ten  tens  of  thousands,  make  one 
of  the  order  of  hundreds  of  thousands ;  ten  hundreds  of 
thousands,  make  one  of  the  order  of  millions,  &c. 

Wherever  there  are  nine  units  of  any  order,  if  there  is 
another  added,  the  number  becomes  one  unit  of  the  next 
higher  order. 

If  we  had  nine  cents,  and  should  add  another,  instead  of 
calling  the  amount  ten  cents,  we  could  call  it  one  dime  ; 
and  so  when  ten  units  are  added  together,  we  can  call 
them  one  unit  of  the  order  of  tens,  instead  often  units  of  the 
unit  order  ;  and  when  we  have  ten  units  of  the  order  of 
fens,  we  can  call  them  one  unit  of  the  order  of  hundreds. 

Questions. — If  nine  cents  have  one  more  added,  in 
what  order  do  they  become  a  unit  ? 

If  nine  dimes  have  another  added,  in  what  order  do 
they  become  units  ? 

Ten  units  of  the  order  of  dollars,  make  one  unitof  what 
order  ? 

Ten  tens,  make  one  unit  of  what  order  ? 

Ten  units,  make  one  unit  of  what  order  ? 

Ten  hundreds  make  one  unit  of  what  order  ? 

The  following  are  the  names  of  the  orders. 
First  order.  Units. 


NUMERATION. 


47 


Second  order, 
Third  order, 
Fourth  order, 
Fifth  order. 
Sixth  order, 
Seventh  order, 
Eighth  order. 
Ninth  order, 
Tenth  order, 
Eleventh  order, 
Twelfth  order, 
Thirteenth  order, 
Fourteenth  order. 
Fifteenth  order. 
Sixteenth  order. 
Seventeenth  order. 
Eighteenth  order, 
Nineteenth  order. 
Twentieth  order, 
Twenty-first  order, 


Tens. 

Hundreds. 

Thousands. 

Tens  of  thousands. 

Hundreds  of  thousands. 

MiUions. 

Tens  of  milHons. 

Hundreds  of  millions. 

Billions. 

Tens  of  billions. 

Hundreds  of  billions. 

Trillions. 

Tens  of  trillions. 

Hundreds  of  Trillions. 

Quadrillions. 

Tens  of  Quadrillions, 

Hundreds  of  Quadrillions. 

Quintillions. 

Tens  of  Quintillions. 


Hundreds  of  Quintillions. 
Twenty-second  order,    Sextillions. 

Sextiliions  are  as  high  as  there  is  ordinarily  any  need 
of  writing  or  reading. 

In  all  the  above  orders,  "  Ten  units  of  one  order,  make 
one  unit  of  the  next  higher  order. 

If  a  figure  2  stands  in  the  Jirst  order,  what  number  does 
it  express?  What  number  does  it  express,  if  it  stands  in 
the /b?/rt/i  order?  In  the  *ecora(i order ?  In  the  ^iA  or- 
der ?     In  the  sixtli  ?  seventh  ?  eighth  1 

Let  the  pupil  write  the  following  : 


1.  Five  units. 

2.  Three  tens  ;  two  units. 

3.  Thirty.two. 

4.  Tluee  and  ten,  or  thirU.  v?. 

5.  Four  and  ten. 

6.  Four  tens,  or  forty. 

7.  Six  and  ten. 
S.  Six  tens. 

9.  Sixteen. 

10.  Si.xty. 

11.  One  hundred  and  sixteen. 

12.  One  hundred,  one  ten,  and  six. 


I    13.   One  hundred  and  sixty. 

14.  One  hundred,  and  six  tens. 

15.  Twoiiundred,  two  tens. 

16.  Two  hundred  and  twenty. 

17.  Two  hundred  and  thirty. 

18.  Two  tens  and  two  units. 

19.  Twenty-two. 

'20.  Two  hundreds  and  two  units. 

21.  Five  tens  and  two  units. 

22.  Five  hundreds. 

23.  Five  tens. 

24.  Fifty. 


48  ARITHMETIC.       SECOND    PART. 


25.  Five  hundred,  and  five  units. 

26.  Five  and  ten. 

27.  Fifteen. 

28.  Fifty  seven. 

29.  Four  hundreds,  six  tens. 

30.  Pour  liundred  and  sixteen. 

31.  Four  hundreds,  one  ten,  and  six. 

32.  Foui'  hundred,  and  six. 


33.  Two  hundred  and  sixty-six. 

34.  Three  hundred,  ten,  and  one. 

35.  Three  hundred  and  eleven. 

36.  Three  hundred,  ten  and  two. 

37.  Three  hundred  and  twelve. 

38.  Four  hundred  and  one. 

39.  One  hundred  and  forty-two. 

40.  Two  hundreds,  two  tens. 


Let  the  pupil  write  the  following  : 

1.  One  unit  of  the  fourth  order.  What  number  is  it  ? 
Which  orders  have  ciphers  in  them  ? 

2.  Two  units  of  the  fourth  order ;  one  unit  of  the  sec- 
ond order,  and  one  unit  of  the  first  order.  What  number 
is  it  ?     What  order  has  a  cipher  in  it  ? 

3.  Two  thousands  ;  one  hundred  ;  five  tens  ;  six  units. 

4.  Twenty-one  hundreds  ;  five  tens  ;  six  units. 

Is  there  any  difference  between  the  two  last  numbers  ? 

5.  Three  thousands,  four  hundreds,  six  tens  and  three 
units. 

G.  Thirty-four  hundred,  and  sixty. three. 

Is  there  any  difierence  in  the  two  last  numbers  ? 

7.  Three  thousands  and  three  units. 
Which  orders  have  ciphei's  placed  in  them  ? 

8.  Three  thousands,  six  hundreds. 
Which  orders  have  ciphers  placed  in  them  ? 

9.  Thirty-six  hundred. 

What  two  ways  of  reading  this  last  number  ? 

10.  Twenty  thousand. 

11.  Two  tens  of  thousands. 

Is  there  any  difference  between  these  two  last  num- 
bers ? 

12.  Twentyifour  thousand. 

What  two  ways  of  reading  this  last  number? 

13.  One  hundred  thousand,  two  tens  of  thousands,  five 
thousands,  six  hundreds,  four  tens,  and  three  units. 

14.  One  hundred  and  twenty-five  thousand,  six  hun- 
idred  and  forty-three. 

Is  there  any  difierence  between  these  two  last  numbers  ? 

15.  Two  tens  of  thousands,  one  thousand,  four  hun- 
dreds, six  tens,  five  units. 

What  two  ways  of  reading  this  number  ? 

16.  Four  hundred  and  sixty-two  thousand,  five  hundred 
and  six. 


NUMERATION.  49 

What  two  ways  of  reading  this  last  ? 

17.  Forty-four  thousand,  tour  liundred  and  forty-four. 
What  two  ways  of  reading  this  last  ? 

18.  Four  hundreds  of  thousands,  five  thousands,  six 
hundreds,  two  tens,  five  units. 

What  two  ways  of  reading  this  last  numher? 

19.  Two  hundred  thousand,  two  thousand,  two  units. 
What  orders  have  cyphers  placed  in  them  ? 

20.  Twenty  thousand,  and  two  units. 

21.  Two  hundred  and  six  thousands,  four  hundred  and 
six. 

22.  Sixty-four  thousand  and  three. 

23.  Sixteen  thousand, 

24.  Fourteen  thousand  and  seven. 

25.  Five  tens  of  thousands,  and  six  units. 

26.  Two  hundreds  of  thousands,  two  hundreds,  two 
units. 

27.  Two  hundred  and  sixty-four  thousand,  and  six. 

28.  Four  thousand,  and  five  units. 

29.  One  hundred  thousand,  and  three. 

30.  Sixteen  thousand,  six  hundred  and  six. 

31.  Twenty-four  tliousand  and  three. 

In  order  to  read  and  write  numbers  more  conveniently, 
they  are  divided  into  periods  of  three  figures  each,  by 
means  of  commas,  thus  : 

87(J,4G9,764,256,622,895,946,852. 

l^hefirst  right  hand  period  is  called  ihe  unit  period  ;  and 
contains  the  orders  caWcd  U".ifs,  tens  and  hundreds 

The  second  period,  is  called  the  thousand  period ;  and 
contains  the  orders  caWed  thousands,  tens  of  thousands,  and 
hundreds  of  thousands. 

The  third  period  is  called  the  tnillion  period,  and  con 
tains  \h a  07-ders  called  millions,  tens  of  millions,  and  hun 
dreds  oj  millions. 

The  fourth  period  is  called  the  billion  period ;  and  con 
tains  the  orders  called  billions,  tens  of  billions,  and  hun 
dreds  oJ  billions. 

The  ffth  period  is  called  the  trillion  period;  and  con 
tains  the  orders  called  trillions,  tens  of  trillions,  and  hun 
dreds  of  trillions. 

5 


50  ARITHMETIC.       SECOND  PART. 

The  sixth  period  is  called  the  quadrillion  period  ;  and 
contains  the  orders  called  quadrillions,  tens  of  quadrillions, 
and  hundreds  of  quadrillions. 

The  seventh  period  is  called  the  quintillion  period  ;  and 
contains  the  orders  called  quintillions,  tens  of  quintillions, 
and  hundreds  of  quintillions. 

The  eighth  period  is  the  sextillion. 

The  Ibllowing  are  the  periods  which  must  be  learned 
in  succession,  beginning  with  the  highest,  as  well  as  with 
the  lowest ;  thus, 


First  Period    Unit. 
Second  Period,  Thousand. 
Third  Period,  Million. 
Fourth  Period,  Billion. 
Fifth  Period,  Trillion. 
Sixth  Period,  Quadrillion. 


Eighth  Period,  Sextillion. 
Seventh  Period,  Quintillion. 
Sixth  Period,  Quadrillion. 
Fifth  Period,  Trillion. 
Fourth  Period,  Billion. 
Third  Period,  Million. 


Seventh  Period,  Quintillion. ^Second  Period,  Thousand. 
Eighth  Period,  Sextillion.     [First  Period,  Unit. 

What  is  the  first  period  1  the  third  ?  the  fifth  ?  the 
eecond  ?  the  fourth  ?  the  seventh  ?  the  sixth  ?  the 
eighth  ? 

The  pupil  may  write  the  naaies  over  the  periods  until 
accustomed  to  reading  them  ;  thus. 

Trill.         Bil.         Mil.         Thous.         Units. 
32  427         983  254  693 

The  above  may  be  read  in  the  following  manner  : 

The  first  left  hand  period  is  read,  3  tens  of  trillions  ;  2 
units  of  trillions  :  ov  thirty -two  triUions. 

The  next  period  is  read,  4  hundreds  of  billions  ;  2  tens 
of  billions  ;  7  units  of  billions  ;  ov  four  hundred  and  twen- 
ty-seven hillions. 

The  next  period  is  read,  i)  hundreds  of  millions  ;  8  tens 
of  milfions  ;  3  units  of  millions,  or  nine  Jnmdi'ed  and  eigh- 
ty-three millions. 

The  next  period  is  read,  2  hundreds  of  thousands  ;  5 
tens  of  thousands  ;  4  units  of  thousands  ;  or  two  hundred 
and  fifty-four  thousand. 

The  next  period  is  read,  6  hundreds  ;  9  tens  ;  3  units  ; 
or  six  hundred  and  ninety-three. 

The  following  is  a  number  in  which  several  orders  are 
omitted,  having  ciphers  in  place  of  numbers. 


NUMERATION. 


51 


Quin.       Quad.       Tril.       Bil.       Mil.       Tli.        U. 
33  067         004       803       064        000      400 

Let  the  pupil  first  tell  what  'periods  and  what  orders  are 
omitted,  having  ciphers  instead  of  numbers. 
The  above  number  may  be  read  thus  : 
Begin  at  the  left  and  read  ;  3  tens  of  quintillions,  and 
3  units  of  quintillions  ;  or  thirty  three  quiiif  ill  ions. 

The  next  period  is,  no  hundreds  of  quadrillions  ;  6  tens 
of  quadrillions  ;  and  seven  units  of  quadrillions  ;  or  sixty- 
seven  quadrillions. 

The  next  period  is,  no  hundreds  of  trillions  ;  no  tens  of 
trillions;  4  units  of  trillions  ;  ov  four  trillions. 

The  next  period  is,  8  hundreds  of  billions;  no  tens  of 
billions ;  3  units  of  billions  ;  or  eight  hundred  and  three 
billions. 

The  next  period  is,  no  hundreds  of  millions  ;  6  tens  of 
millions  ;  4  units  of  millions  ;  or  sixty-Jour  millions. 

The  next  period,  as  it  has  no  hundreds,  tens,  or  units 
of  thousands,  may  lie  omitted  entirely,  when  reading. 

The  next  period  is,  4  hundreds  ;  no  tens  ;  no  units  ;  or 
four  hundred. 

The  best  and  most  common  way  of  reading,  is  that  in 
the  italics,  and  then  all  together,  it  reads  thus  : 

Thirty-three  quintillion  ;  sixty-seven  quadrillion  ;  four 
trillion  ;  eight  hundred  and  three  billion  ;  sixty-four  mil- 
lion ;  four  hundred. 

Let  the  pupil  read  the  following  sum  in  both  ways  : 
Quin.      Quad.      Tril.      Bil.       Mil.      Th.      Un. 
607        300  000      763      490      068      002 


KULE-  FOR    READING    WHOLE    NTTMBERS. 

Point  off  into  periods  of  three  figures  each,  beginning  at 
the  right.  Read  each  period  as  if  it  stood  alone,  and  then 
add  the  name  of  the  period. 

Note. — When  a  period  or  order  is  omitted,  it  is  not  ne- 
cessary  to  mention  it  at  all. 

Before  reading,  let  the  pupil  tell  what  periods  and  or- 
ders are  omitted,  and  represented  by  ciphers. 


52 


2 

ARITHMETIC. 

SECOND    PART. 

Let  tlie  pupil  point  off,  and  read  the  following  figures 

1 

2 

31 

304 

300046 

200200200 

111 

24 

40 

600 

300005 

2030003000 

100 

136 

400 

611 

1200437 

311001300 

101 

3024 

4040 

693 

1200039 

60009090 

1011 

2002 

6000 

4004 

4960004 

100100001 

2002 

46900 

40640 

103006 

1430096 

2071113603 

3041 

60021 

600003 

1063007 

6000007 

1000673 

201 

62003 

100014 

103964 

86004369 

101700013 

9010 

6040064 

600436 

140001 

20064000 

600040006 

3004 

46923 

64003 

400006 

400400400 

300010000 

227034293 

9623000062 

200001900 

10043259054 

3690200000 

43600078609 

30006340200 

459643723007 

602030004296 

612942004000040367 

40000643209437 

3907650060042300000 

237 

6000964 

300600C 

)0 

396770000543965000076 

It  is  necessary  for  the  pupil  to  understand,  that  the 
French  and  English  arithmeticians  use  different  methods 
of  numeration. 

The  English  have  their  periods  contain  six  orders,  and 
the  French  only  th'ee. 

This  makes  no  difference  till  we  come  to  hundreds  of 
millions.  After  that,  it  makes  a  great  difference,  as  will 
be  seen  by  the  following  comparison. 

It  must  be  noticed,  that  the  same  figures  are  used  in  both. 


English  Method. 


Trillions. 
579364, 


Billions. 
028635, 


Millions. 
419763, 


Units. 
215468. 


French  Method. 


Sext.  Quin.  Qua.  Trill.  Bill.  Mill.  Th.  Units. 
579,       364,      028,      635,      419,     763,      215,      468. 

From  the  above  it  can  be  seen,  that  all  the  orders  above 
hundreds  of  millions,  in  both  methods,  give  the  smne  name, 
to  a  very  different  value. 

Thus,  the  orders  of  thousands  of  millions,  tens  of  thous- 
ands of  millions,  and  hundreds  of  thousands  of  millions,  in 
the  English  method,  would  be  read  as  billions,  tens  of  bill- 
ions and  hundreds  of  billions,  in  the  French  method. 


NUMERATION. 


53 


Billions,  tens  of  billions,  and  hundreds  of  billions,  in  the 
English  method,  are  equivalent  to  trillions,  tens  of  trillions, 
and  hundreds  of  trillions,  in  the  French  method. 

Five  trillion,  in  the  French  method,  would  be  read  fire 
billion,  in  the  English  ;  and  five  trillion,  in  the  English 
method,  would  be  read  five  qaadrillion,  in  the  French. 

Questions. — How  would  a  billion,  in  the  English  meth- 
od,  be  read  in  the  French? 

How  would  one  hundred  billion,  in  the  English  method, 
be  read  in  the  French  ? 

How  would  one  billion,  in  the  French  method,  be  read 
in  the  English  ? 

How  would  six  hundred  billion  in  the  French  method, 
be  read  in  the  English  ? 

The  French  method  is  adopted  in  this  work,  because 
it  is  both  the  most  convenient,  and  the  most  common. 
But  the  pupil  needs  to  understand  the  difference  be- 
tween the  two  modes,  and  the  teacher  should  make  the 
class  point  off  and  read  numbers  by  both. 

Point  off  and  read  the  following  numbers,  first  by  the 

French,  and  then  by  the  English  method. 

7G543217G50U431 

9870000(554321765432 

326980000000400003G0093 

436789643645904379029364 

In  order  to  write  numbers  correctly,  the  pupil  must 
learn  thoroughly,  the  succession  of  the  orders  beginning 
at  the  left.  Thus,  SextiUion,  Quintillion,  Quadrillion, 
Trillion,  Billion,  Thousand  and  Unit. 


Rule  for  Wkiting  Whole  Numbeks. 

Begin  with  the  higliest  period,  and  write  first  the Jiund reds, 
then  the  tens,  and  then  the  wuts  of  that  period.  Proceed  thus, 
until  all  the  periods  are  written.  Place  a  comma  between 
each  period.  If  any  period  or  order  is  omitted,  place  ciphers 
in  their  place. 

Note. — Ciphers  prefixed  to  a  whole  mimber,  have  no 
effect  upon  the  value.  A  number,  therefore,  should  nev- 
er  be  begun  with  a  cipher. 

5* 


54  ARITHMETIC.       SECOND  PART. 

Write  two  thousand  and  two.  What  orders  are  omit. 
ted? 

Write  two  million,  two  thousand,  and  four.  What  or- 
ders are  omitted  ? 

Write  Three  hundred  and  twenty-four.  What  period 
and  orders  in  this  number  ? 

Write  Two  hundred  thousand  and  four.  What  orders 
omitted  in  ihis  last  number  ? 

Write  ;  Two  million  and  six  ?  What  period  omitted  ? 
what  oi'ders  omitted  ? 

Write,  Six  million  ;  two  hundred  and  three.  Which 
period  and  what  orders  are  omitted  ? 

Write,  Twenty. four  million  ;  three  hundred.  Which 
period  and  what  orders  are  omitted  ? 

Write  tiie  lollowing  sums  and  mention  the  periods  and 
orders  which  are  omitled. 

1.  One  billion;  twenty-four  million;  three  thousand 
and  three. 

2.  Four  hundred  and  sixty-nine  billion ;  forty-four 
thousand ;  and  seventeen. 

3.  Fifty  billion  ;  three  hundred  million  ;  four  hundred 
and  fifty  thousand  ;  and  nineteen. 

4.  Fifty  billion,  and  seven. 

5..  Four  hundred  and  thirteen  million,  and  two  thous- 
Rnd. 

6.  Nineteen  billion,  and  one  million. 

7.  Six  trillion  ;  nine  thousand,  and  ten. 

8.  Seven  trillion  ;  nineteen  billion  ;  ten  thousand,  and 
four  hundred. 

9.  Four  hundred  and  nine  trillion  ;  Sixteen  million ; 
eleven  thousand  and  forty. 

10.  Fifteen  billion  ;  two  hundred  and  four  million  ;  six 
thousand,  and  twent3f-one. 

!1.  Sixty-four  million ;  "four  hundred  thousand  ;  three 
hundred. 

12.  SixteeH  million  ;  five  hundred  thousand,  and  six. 

13.  Three  trillion  ;  fourteen  million  ;  seven  thousand. 

14.  Two  hundred  and  sixteen  million. 

15.  Two  billion  ;  sixteen  million,  and  sixteen. 

16.  Three  hundred  and  six  trillion  ;  four  thousand,  and 
six. 


NUMERATION.  55 

i7.  Two  quintillion  ;  six  quadrillion  and  five. 

18.  Three  hundred  and  sixty-four  thousand. 

19.  Tluee  nnillion  and  six. 

20.  Fourteen  trilUon  ;  three  hundred. 

21.  Sixteen  trilhon,  four  milHon,  two  hundred  and  four 
thousand,  seven  hundred  and  one. 

2"i.  Three  sextilHon,  one  hundred  quadrilhon,  fourteen 
trillion,  two  hundred  and  sixty  billion,  four  hundred  mill- 
ions,  sixteen  thousand,  four  hundred  and  one. 

23.  Five  million,  two  hundred  thousands,  and  sixty-two. 

24.  Two  hundred  and  five  mdlions,  and  seventy-four. 

25.  Twelve  hundred  and  six  billions,  four  millions,  and 
six  thousand. 

26.  Two  hundred  sextillions,  four  hundred  millions, 
three  hundred  and  four  thousand,  two  hundred  and  six. 

27.  Fifteen  quintillion,  six  quadrillions,  one  hundred 
trillions,  forty-four  billions,  two  nullions,  and  forty-nine. 

28.  Fifty  quadrillions,  six  hundred  trillions,  forty-three 
millions,  two  thousands  four  hundred  and  six. 

29.  Two  hundred  and  six  trillions,  forty-three  billions, 
four  hundred  and  nine  millions,  sixty-four  thousands,  four 
hundred  and  ninety-six. 

30.  One  hundred  and  four  billions,  six  millions,  forty- 
nine  t!i()usands,  four  hundred  and  ninety -six. 

31.  Thirteen  nullions,  four  hundred  thou.sands,  six  hun- 
dred and  forty-nine. 

32.  Six  sextUlii'ns,  five  quintillions,  four  quadrillions, 
three  trillions,  two  billions,  and  one  mdlion. 


NUMERATION  OF  VULGAR  FRACTIONS. 

Figures  are  of  two  kinds,— Figures  for  a  number  of 
whole  things,  and  figures  for  a  number  of  parts  of  things. 

A  unit  is  a  whole  thing  of  any  kind. 

A  fraction  is  a  part  of  one  thing ;  or  a  part  of  several 
things. 

Figures  may  therefore  be  divided  into  fractional  and 
unit  figures. 

The  following  is  the  mode  of  showing  when  the  nam- 


56  ARITHMETIC.       SECOND  PART. 

bers  represented  are  several  whole  things,  and  when  they 
are  several  parts  of  things. 

When  there  are  two  whole  things,  their  number  is  ex- 
pressed thus,  (2).     This  is  called  a  unit  figure. 

But  if  a  whole  thing  is  divided  into  three  parts,  and  we 
wish  to  express  two  of  these,  by  figures,  we  write  one  fig- 
ure, to  show  into  how  many  parts  the  whole  thing  is  divi- 
ded, and  then  above  it,  write  the  number  of  parts  we  wish 
to  express  ;  thus,  (|). 

This  is  called  a  fractional  figure.  The  lower  figure 
shows  into  how  many  parts  the  whole  thing  is  divided, 
and  the  ttjiper  figure  shows  how  many  of  these  parts  are 
expressed. 

In  |,  into  how  many  parts  is  the  whole  thing  divided, 
and  how  many  of  these  parts  are  expressed? 

In  A,  into  how  many  parts  is  the  whole  thing  divided, 
and  how  manv  parts  are  expressed  ? 

In  f  ?  In  f  ?  In  ^  ?  In  f  ?  In  j\1  In  ^\  ? 
Fractional  figures  show  into  how  many  parts  one  whole 
thing  is  divided,  and  how  many  of  these  parts  are  expres- 
sed. Besides  this,  they  can  show  what  part  is  taken  from 
several  whole  things.  Thus  |  shows  that  one  thing  is  divi- 
ded into  four  parts,  and  three  of  them  are  taken  ;  or  that 
three  whole  things,  have  a  fourth  taken  from  each  of  them. 
For,  three  fourths  of  one  whole  thing,  is  the  same  quantity 
as  one  fourth  of  three  whole  things. 

If  you  have  three  apples,  and  take  07ie  fourth  out  of 
each,  how  much  will  you  have,  and  how  will  you  express 
it  in  figures  1  If  you  divide  one  apple  into  Jour  parts,  and 
take  three  of  these  parts,  how  do  you  express  the  quantity 
taken  ? 

If  you  have  two  apples,  and  take  one  sixth  from  each, 
how  much  will  you  have,  and  how  will  you  express  it  in 
figures  ? 

If  you  divide  an  apple  into  six  parts,  and  take  two  of 
these  "parts,  how  much  will  you  have,  and  how  will  you 
express  it  in  figures  ? 

If  an  apple  is  divided  into  eight  parts,  and  you  take 
three  of  them,  how  much  will  you  have,  and  how  will  you 
express  it  in  figures  ? 

If  the  fraction  is  considered  as  showing  how  many  parts 


NUMERATION    OF    VULGAR   FRACTIONS.  57 

are  taken  from  one  unit,  then  the  lower  figure  shows  into 
hoiv  many  j>aris  a  unit  is  divided,  and  the  uj)pcr  figure 
shows  how  many  of  these  parts  are  taken.  liut  il'the  I'rac- 
tion  is  considered  as  showing  what  part  is  taken  out  of 
sexieral  units,  then  the  upper  figure  shows  the  number  of 
unrls,  and  the  lower  figure  shows  what  part  is  taken  from 
each. 

Thus  the  fraction  |  may  be  considered  as  expressing, 
two  sixths  of  one  thing,  or  as  one  sixth  of  two  things. 

•j\  is  either  one  twelfth  of  thixe  things,  or  tJiree  twelfths 
of  one  thing. 

^  is  either  four  ffths  of  one  thing,  or  one  fifth  of  four 
things. 

f  either  shows  that  one  ninth  is  taken  out  of  two  things  ; 
or  that  two  ninths  are  taken  out  of  one  thing. 

If  I  is  considered  as  showing  how  many  parts  are  taken 
out  of  one  thing  it  is  four  sevenths  of  one  unit.  If  it  is  con- 
sidered as  showing  what  part  is  taken  out  of  several 
things,  it  is  one  seventh  of  four  units. 

If  I  shows  how  7nany  j}arts  are  laken  out  ofo>ic  thing,  it 
is  two  thirds  of  one  thing.  If  it  shows  w hat  part  is  taken 
out  of  several  things,  it  is  one  third  of  two  things. 

If  ^  is  considered  as  showing  how  many  parts  are  taken 
out  of  oHe  unit,   what  does  the  8  sliow,  and  wiiat  does  the 

7  show? 

If  it  is  considered  as  expressing  what  part  is  taken  out 
of  several  units,  what  does  the  7  show,  and  what  do.es  the 

8  show  ? 

If  *  is  considered  as  expressing  how  many  parts  are  ta- 
ken out  of  one  unit,  what  does  the  6  show,  and  what  does 
tlie  4  show  ?  If  it  is  considered  as  expressing  what  part 
is  taken  out  of  several  units,  what  does  the  4  show,  and 
what  does  the  6  show  1 

Whenever  the  numerator  is  larger  than  the  denominator, 
the  fraction  is  called  an  improper  fraction,  and  always  is 
to  he  considered  as  expressing  what  part  is  taken  out  of 
several  units. 

Which  of  the  following  are  improper  fractions  ? 

3* 


What  does  an  improper  fraction  show  ? 


58 


ARITHMETIC.       SECOND    PART. 


RULK    FOR    READING    VULGAR    FRACTIONS. 

Read  the  number  of  parts  expressed  btj  the  numerator,  and 
then  the  size  oftheparts  expressed  by  the  denominator  ;  or 

Read  the  part  expressed  by  the  denominator,  and  then  the 
number  of  %m.its,  expressed  by  the  numerator. 

Read  the  following  fractions  in  both  ways,  thus  :  f  is 
either  three  fourths  of  one  thing,  or  one  fourth  of  three 
things. 

f  is  either  three  fifths,  or  one  fifth  of  three. 


RULE  FOR  WRITING  VULGAR  FRACTIONS. 

Write  the  number  of  parts  into  which  a  unit  is  divided, 
and  draw  a  line  above  it.  Over  it  torite  the  number  of 
parts  which  are  to  be  expressed;  or 

Write  the  whole  nuinbers  which  have  a  certain  part  taken 
from  them,  and  draw  a  line  wider.  Beneath  it  write  the 
figure  lohich  expresses  the  part  which  is  to  he  taken  out  of 
each  of  the  units  above. 

Let  the  pupil  write  the  following  : 

If  a  man  divided  an  apple  into  eight  parts,  and  gave 
away  five  of  these  parts,  how  do  you  express  the  quantity 
he  gave  away,  and  the  quantity  he  kept? 

If  a  man  had  three  apples,  and  cut  out  a  fourth  part  of 
each,  and  gave  it  away,  how  do  you  express  what  he  gave 
away  X 

If  a  man  had  twelve  oranges,  and  one  sixth  of  each  was 
decayed,  how  do  you  express  the  quantity  of  decayed  or- 
anges he  had  1 

If  a  man  had  five  casks  of  wine,  and  a  twelfth  part  leak, 
ed  out  of  each,  how  do  you  express  what  he  lost  ? 


DECIMAL  NUMERATION. 

There  is  another  mode  of  writing  fractions,  in  which  the 
numerator  only,  is  written.  The  denominator,  although 
not  written,  is  always  understood  to  be  1,  and  a  certain 
number  of  ciphers. 

These  fractions  are  called  Decimals. 


NUMERATION    OP    VULGAR    FRACTIONS.  59 

Thus  in  writing  decimals,  if  we  are  to  express  twotenihs, 
instead  of  writing  it  thus  -f^,  the  numerator  only  is  written, 
and  a  comma,  called  a  separatrix,  is  placed  before  it, 
thus  ,2. 

The  following  is  the  rule,  by  which  it  is  known  what  is 
the  denominator. 

The  denominator  of  a  decimal  is  always  1 ,  nnd  as  many 
cyphers  as  there  are  figures  in  the  numerator,  or  decimal. 

What  is  the  denominator  of  this  decimal,  ,2  1 

Ans.     1  and  one  cjpher. 

How  many  cyphers  in  the  denominators  of  these  deci- 
mals, ,34.   ,<306.  ,3246.  ,56945.  ,3694.  ? 

If  the  decimal  has  one  figure,  it  expresses  tenths.  Thus 
,2  is  two  tenths. 

If  it  has /wo  figures,  it  QX^xes&QS  hundredths.  Thus  ,02 
is  two  hundredths. 

If  it  has  //tree  figures,  it  expresses  thousandtJis.  Thus 
,002  is  two  thousandths. 

If  it  has  four  figures,  it  expresses  tens  of  thousandths. 
Thus  ,0002  is  two  tens  of  thousandths. 

If  it  has  fiie  figures,  it  expresses  hundreds  of  thous- 
andths     Thus  ,00002  is  two  hundreds  of  thousandths. 

What  does  this  decimal  express,  ,3? 

Ans.     Three  tenths. 

What  does  this  decimal  express,  ,30  ? 

Ans.     Tliirty  hundredtlis. 

What  does  this  decimal  express,  ,003? 

Ans.     Three  thousandths. 

What  does  this  decimal  express,  ,0003  ? 

Ans.     Three  tens  of  thousandths. 

What  does  this  decimal  express,  ,5  1     Ans.  5  tentlis. 

What  does  this  decimal  express,  ,15  ?  Ans.  Fifteen 
hundredtlis. 

What  does  this  decimal  express,  ,110? 

What  does  this  decimal  express,  ,2000  ? 

What  docs  this  decimal  express,  ,00002? 

Ans.     Two  hundredths  of  thousandths. 

A  decimal  must  always  have  the  number  of  figures  in 
the  numerator,  equal  to  the  number  of  ciphers  in  the  de- 
nominator ;  therefore  it  is  necessary  to  learn  how  many 
ciphers  there  are  in  each  kind  of  denominator. 


60  ARITHMETIC.       SECOND  PART. 

If  the  decimal  is  tenths,  there  is  one  cipher  in  the  de- 
nominator ;  if  hvndredihs,  there  are  two  ciphers ;  if 
thousandths,  there  are  three  ciphers  ;  if  tens  of  thous- 
andths, there  are  four  ciphers ;  if  hundreds  of  thous- 
andths, there  are  five  ciphers,  &c. 

Of  course  in  writing  decimals,  if  tenths  are  to  be  ex- 
pressed,  there  must  be  only  one  figure  in  the  mimerator,  or 
decimal  ;  if  hundredths,  there  must  be  two  figures  ;  if 
thousandths,  there  must  be  three  figures;  \{ tens  ofthous- 
andths,  there  must  he  four  figures  ;  i?  hundreds  of  thous- 
andths, there  must  hefve  figures,  &c. 

If  you  are  to  write  two  tenths,  how  many  figures  must 
there  be  in  the  numerator  or  decimal,  and  how  many  ci- 
phers are  understood  to  be  in  the  denominator  ?  Write 
two  tenths.     (,2). 

If  you  are  to  write  two  hundredths  how  many  ciphers 
are  understood  to  be  in  the  denominator,  and  how  many 
figures  must  there  be  in  the  numerator  ? 

Write,  two  hundredths. 

In  writing  this  last,  the  pupil  must  first  write  the  2,  and 
then  as  there  must  be  as  many  figures  in  the  numerator, 
as  there  are  ciphers  in  the  denominator,  a  cipher  is  placed 
before  the  2,  and  then  the  separatrix  is  prefixed  thus,  ,02. 

If  the  cipher  were  placed  ajtcr  the  2,  how  would  it 
read  ? 

Ans.     Twenty  hundredths,  instead  of  two  hundredths. 

If  the  cipher  were  not  placed  before  the  2,  how  would 
it  read  ?     Ans.     Two  tenths. 

If  another  cipher  is  placed  before  the,02thus,  ,002  how 
does  it  read  1 

What  does  the.  denominator  express,  when  there  are 
three  figures  in  the  decimal.     Ans.      Thousandths. 

What  does  it  express  when  there  are  four  figures  in  the 
decimal  ? 

Let  the  pupil  write  the  following. 

1 .  Two  tenths. 

2.  Two  hundredths. 

3.  Two  thousandths. 

4.  Two  tens  of  thousandths. 

'   5.  Two  hundreds  of  thousandths. 
6.  Five  tenths. 


DECIMAL    NUMERATION.  61 

7.  Fifteen  hundredths. 

8.  Fifteen  thousandths. 

9.  Fifteen  tens  of  thousandths. 

10.  Fifteen  hundreds  of  thousandtlis. 

11.  One  tenth. 

12.  Eleven  hundredths. 

13.  One  hundred  and  fifteen  thotisandths. 

14.  Five  tenths. 

15.  Fifty-five  hundredths. 

16.  Five  hundred  thousandths. 

17.  Five  hundred  and  five  thousandths. 

18.  Fifteen  thousandths. 

19.  Five  thousandths. 

20.  Two  hundred  thousandths. 

21.  Twenty-nine  thousandths. 

22.  Five  hundredths. 

23.  Forty  hundredths. 

24.  Nine  fe?<s  o/*  thousandths. 

25.  Nineteen  fez/s  of  thousandths. 

26.  Nine  hundred  tens  of  thousandths. 

27.  Two  thousand  <f'»K<f  of  thousandths. 

28.  Two  thousand  and  two  tens  of  thousandths. 

29.  Three  thousand  three  hundred  /???«  o/  thousandths. 

30.  Thirty-two  hundred  <fH.?  o/"  thousandths. 

31.  S\x  tens  of  thousandths. 

32.  Four  hundreds  of  thousandths. 

33.  Fourteen  hundreds  of  thousandths. 

34.  Four  hundred  hundreds  of  thousandths. 

35.  .Two  thousand  and  six  hundreds  of  thousandths. 

36.  Sixty-four  thousand  hundreds  of  thousandths. 

37.  Sixteen  thousand  and  four  hundreds  of  thousandths . 

38.  Four  thousand  and  nine  hundreds  of  thousandths. 

39.  Six  hundreds  of  thousandths. 

40.  Five  thousand  and  four  hundreds  of  thousandths. 

41.  Sixty-five  thousand  hundreds  of  thousandths. 

42.  Nine  hundred  and  one  hundreds  of  thousandths. 

43.  Twenty-nine  hundred  hundreds  of  thousandths. 

44.  Twelve  tens  of  thousandths. 

45.  Fifteen  hundredths. 

46.  Sixty-four  thousandths. 

47.  Nine  hundred  and  one  tens  of  thousandths. 

6 


62  ARITHMETIC.       SECOND    PART. 

Decimals  can  be  read  in  two  different  ways. 

Thus,  21  can  be  read,  either  as  two  tenths,  and  one 
hundredth  ;  or  as  twenty-one  hundredths. 

This  can  best  be  illustrated,  by  the  coin  of  the  United 
States.  Thus,  2  dimes,  1  cent,  can  be  read,  either  as 
twenty-one  cents,  or  as  two  dimes  and  one  cent. 

Thus  again,  1  dollar,  3  dimes,  and  2  cents,  can  be 
called,  either  132  cents  ;  or  13  dimes,  2  cents  ;  or  1  dol- 
lar, 3  dimes,  and  2  cents. 

In  like  manner,  decimals  may  be  read  in  different  ways. 
Thus,  234  can  be  read  either  as  234  thousandths ;  or  2 
tenths,  3  hundredths,  and  4  thousandths  ;  or  23  hun- 
dredths, and  4  thousandths ;  or  2  tenths,  and  34  thou- 
sandths. 

Write  two  tenths. 

Write  twenty  hundredths. 

,2  is  how  many  hundredths  ? 

Ans.  There  are  ten  times  as  many  hundredths  as  there 
are  tenths  in  a  thing.  Therefore  ,2  is  ten  times  as  many 
hundredths,  or  20. 

Is  there  any  difference  in  the  value  of  ,2  and  ,20  ?  What 
is  the  difference  between  them  ? 

Ans.  The  ,20  has  ten  times  7nore  pieces,  and  each 
piece  is  ten  times  smaller  than  the  ,2  ;  but  there  is  no 
difference  in  the  value. 

,3  is  how  many  hundredths  ?  ,4  is  how  many  hun- 
dredths  ? 

,30  is  how  many  tenths  1     ,40  is  how  many  tenths  1 

Write  two  tenths,  and  four  hundredths.  In  this  sum 
how  many  hundredths  ? 

Write  thirty.four  hundredths.  In  this  sum  how  many 
tenths  1 

Write  2  tenths,  6  hundredths,  or  twenty-six  hundredths. 

Write  4  tenths,  9  hundredths,  and  read  it  both  ways. 

Write  6  tenths,  7  hundredths,  five  thousandths,  or  six 
hundred  and  seventy-five  thousandths. 

Write  6  tenths,  4  hundredths,  and  .5  thousandths. 

Write  nine  tenths,  six  hundredths,  and  six  thousandths, 
and  read  them  both  ways. 

Write  seven  tenths,  six  hundredths,  five  thousandths, 
and  nine  tens  of  thousandths,  and  read  them  both  ways. 


DECIMAL    NUMERATION.  63 

Write  nine  tenths,  no  hundredths,  six  thousandths,  no 
tens  of  thousandths,  and  five  hundreds  of  thousandths,  and 
read  it  both  ways. 

Write  six  tenths,  no  hundredths,  no  thousandths,  and 
five  tens  of  thousandths,  and  read  it  both  ways. 

Write  six  thousand  four  hundred  and  thirty-six,  tens  of 
thousandths,  and  tell  how  many  tenths,  hundredths,  and 
thousandths  there  are. 

Write  four  hundred  and  seventy. nine  thousandths,  and 
tell  how  many  tenths,  and  hundredths  there  are. 

Write  five  hundred  and  six  thousandths,  and  tell  how 
many  tenths  there  are. 

Write  five  hundred  and  ninety. six  hundreds  of  thou, 
sandths,  and  read  it  both  ways. 

From  the  above  it  appears,  that  in  decimals,  the  order 
next  to  the  separatrix  is  tenths ;  the  second  order  from  the 
separatrix  is  hundredths  ;  the  third  order  is  thousandths  ; 
the  fourth  order  is  tens  of  thousandths  ;  the  ffth  order  is 
hundreds  of  thousandths,  &c. 

Questions. — In  decimals  what  is  the  first  order,  at  the 
right  of  the  separatrix  1  What  is  the  second  order  ?  What 
is  the  fourth  order  ?     What  is  the  third  ?    the  fifth  ? 

Decimals,  are  oflen  written  with  whole  numbers.  Thus, 
2,5.     36,349. 

Whole  numbers  and  decimals  together,  are  called  mixed 
decimals. 

Write  twenty-four  whole  numbers,  and  twenty-four 
hundredths.  Two  hundred  whole  numbers,  and  five 
tenths.     What  are  the  mixed  decimals  ? 


Rule  for  rkading  decimals. 

Read  the  numerator,  as  if  it  icere  whole  numbers,  and  then 
add  the  name  of  the  denominator  ;  or.  Read  the  number  of 
each  separate  order,  and  follow  it  with  the  name  of  the  order 
in  which  it  stands. 

Read  the  following  decimals  both  ways. 

,11.  ,020.  ,5005.  ,32568.  ,0505.  ,521.  ,43002. 
24,690.  6,40043.  6,4000.  69,9604.  86,0092.  2,002. 
16,00020. 

In  writing  decimals  from  the  dictation  of  the  teacher, 


64  ARITHMETIC.       SECOND  PART. 

the  pupil  needs  to  understand  the  two  methods  very  clearl3\ 

Thus  for  example,  he  may  have  this  decimal,  ,00205, 
dictated  in  two  ways,  viz.  :  205  hundreds  of  thousandths, 
or  2  thousandths,  and  5  hundreds  of  thousandths. 

In  the  first  mode  of  dictation,  he  must  write  the  205  as 
if  it  were  whole  numbers,  and  then  prefix  ciphers  to 
make  the  figures  of  the  numerator  equal  to  the  ciphers  of 
the  denominator. 

In  the  second  mode  of  dictation,  he  must  put  a  cipher 
in  each  order  which  is  not  mentioned ;  viz. :  in  the  orders 
tenths,  hundredths,  and  tens  of  thousandths,  and  a  2 
in  the  order  of  thousandths,  and  a  5  in  the  order  of  huii' 
dreds  of  thousandths. 

Let  the  pupil  write  the  following  in  both  methods  of  dic- 
tation. 

8  hundredths,  6  tens  of  thousandths  ;  or  806  tens  of 
thousandths. 

2  tenths,  4  tens  of  thousandths  ;  or  2004  tens  of  .thou- 
sandths. 

2  thousandths,  5  tens  of  thousandths  ;  or  25  tens  of  thou- 
sandths. 

3  hundredths,  6  thousandths,  5  tens  of  thousandths  ;  or 
365  tens  of  thousandths. 


RULE    FOR   WlilTING    DECIMALS. 

Write  the  numerator  as  if  it  were  whole  numbers,  and 
ilien  prefix  a  separatrix.  If  the  figures  of  the  decimal  do 
not  equal  in  number  the  ciphers  of  the  denominator,  pr^x 
ciphers  to  make  them  equal,  before  placing  the  separatrix ;  or 

Write  each  order  separately,  placing  ciphers  in  the  orders 
onittcd. 

Write  the  following  : 

1.  Two  hundred  and  ten  thousandths. 

2.  Two  tenths,  five  thousandths,  six  tens  of  thousandths. 
Here  the  order  o(  hundredtlis  is  omitted,  and  has  a  cipher 
put  in  it. 

3.  Two  hundred  and  four  hundreds  of  thousandths. 

4.  Two  thousandths  ;  four  hundreds  of  thousandtiis. 
What  orders  are  omitted  .'' 

5.  Sixteen  tens  of  thousandths. 


DECIMAL   NUMERATION. 


65 


6.  One  thousandth,  six  tens  of  thousandths.  What  or- 
Jers  are  omitted  ? 

7.  Four  hundred  and  five  thousandths.  What  orders 
omitted  ? 

8.  Four  tenths,  five  thousandths.  What  orders  are 
omitted  ? 

9.  Three  hundred  and  sixty-five  tens  of  thousandths. 
What  order  lias  a  cipher  placed  in  it  ? 

10-  Four  hundredths,  five  tens  of  thousandths.  What 
orders  are  omitted  ? 

11.  Twenty-six  thousand,  nine  hundred  and  forty-six 
hundreds  of  thousandths. 

12.  Two  tenths,  six  hundredths,  nine  thousandths,  four 
tens  of  thousandths,  six  hundreds  of  thousandths. 

In  mixed  decimals,  it  will  be  seen,  that  the  orders  are 
reckoned  from  the  separatrix,  both  ways. 

Thus  in  98423,46795,  the  first  order  at  the  right  of  the 
separatrix  is  fentlis,  and  [\\efirst  order  at  the  left  is  units. 

What  is  the  second  order  at  the  right,  and  the  second  or. 
dev  at  the  left  of  the  separatrix  ? 

What  is  the  third  order  at  the  right,  and  at  the  left  of 
the  separatrix  ? 

What  is  ihe  fourth  order  at  the  right,  and  at  the  left  of 
the  separatrix  ? 

What  is  the  fifth  order  at  the  right,  and  at  the  left  of  the 
separatrix  ? 

If  you  have  the  decimal  ,2,  and  place  a  cypher  at  tho 
right,  thus  ,20,  what  does  it  become  ?  Is  the  value  alter- 
ed ?     How  is  it  altered  1 

Ans.  The  parts  are  made  te7i  times  smaller,  and  there 
are  ten  times  more  of  them,  so  that  the  value  remains  the 
same. 

If  you  place  a  cypher  at  the  left  of  ,2  thus,  ,02,  what 
does  it  becori'C  ?  How  much  smaller  is  a  hundredth,  than 
a  tenth  ? 

How  much  smaller  does  it  make  a  decimal  to  'prefiit  a 
cipher  to  it  ? 

If  you  put  two  ciphers  at  the  right  of  ,2,  what  effect  is 
produced  ?  If  you  put  them  at  the  left  of  it,  what  effect  is 
produced  ? 

The  following  principle  is  exhibited  above  : 

Ciphers  placed  at  the  right  of  decimals,  change  their 
names  but  not  their  value. 
6* 


66  ARITHMETIC.       SECOND  PART. 

Ciphers  placed  at  the  left  of  decimals,  diminish  their  val- 
ue ten  times,  for  every  cipher  thus  prefixed. 

Prefix  a  cipher  to  ,91  and  read  it.  Annex  a  cipher  to 
,91  and  read  it. 

Prefix  a  cipher  to  ,20  and  read  it.  Annex  a  cipher  to 
,'20  and  read  it. 


SlGKS  AND  AbbRKVIATIONS  USED  IN  ARITHMETIC. 

The  following  signs  are  used  instead  of  the  words  they 
represent. 

4"  signifies  piliis  or  added  to. 
—  signifies  minus  or  lessened  by. 
X    signifies  midti plied  by. 
H-  signifies  divided  by. 
=  signifies  equals. 
E.  signifies  Eagles. 
$    signifies  Dollars. 
d.   signifies  Dimes. 
cts.  signifies  cents. 
m.  signifies  mills. 


ADDITION. 

Addition  is  uniting  several  numbers  in  one. 

There  are  four  different  processes  of  addition. 

The  fi-st  is  Simple  Addition,  in  which  ten  units  of  one 
order  make  one  unit  of  the  next  higher  order.  Thus,  ten 
units  make  one  of  the  order  of  tens — Ten  tens  make  one 
of  the  order  of  hundreds — Ten  hundreds,  m;ike  one  of  the 
order  of  thousands,  &;c. 

The  second  is  Decimal  Addilion,  in  vvhic!i  decimul  frac- 
tions are  added  to  each  other.  Thus,  ,5  ,50  ,.505  are  ad"- 
ded  together. 

The  third  is  Compound  Addition,  in  which  other  num- 
bers besides  ten,  make  units  of  higher  orders.  Thus, 
four  units  of  the  order  of  farthings,  make  one  unit  of  the 
order  of  pence.  Twelve  units  of  the  order  of  pence, 
make  one  of  the  shiUing  order.     Twenty  of  the  shilling 


SIMPLE    ADDITION.  67 

order,  make  one  of  the  pound  order,  &c. 

The Jourlh  is  the  Addition  of  Vulgar  Fractions,  in  which 
parts  of  units  are  added  to  each  other.  Thus  i  |  and  | 
are  added  to  each  other. 


SIMPLE  ADDITION. 

If  8  units  are  added  to  9  units,  how  many  are  there 
of  the  order  of  tens  ? 

Write  the  8  under  the  9,  and  draw  a  line  under.  Place 
the  units  of  the  answer,  under  the  figures  added,  and  set 
the  1  ten  hefore  them. 

If  13  apples  are  added  to  25  apples,  how  many  are 
there  in  the  whole  ? 

Write  the  units  under  units,  and  tens  under  tens.  Add 
the  units  first,  and  place  the  answer  under  the  unit  column. 
Then  add  the  tens  in  the  same  way. 

Add  12  cents  to  5  cents. 

Add  13  apples  to  14  apples. 

Add  14  dollars  to  19  dollars. 

Add  5  and  2  and  12  together. 

Add  V.i  and  12  and  14  together. 

Let  the  pupil  add  small  sums,  which  do  not  amount  to 
ten  of  any  order,  till  it  can  be  done  quickly  and  with  a  full 
understanding  of  the  process. 

In  the  next  process  let  the  coins  be  used  to  illustrate. 

If  25  cents  be  added  to  16  cents,  how  many  cents  are 
there  ? 

Let  2  dimes  be  laid  on  the  table,  and  5  cents  placed 
at  the  right  of  them.  Under  the  2  dimes  place  I  dime, 
and  under  the  5  cents  |)lace  G  cents.  Let  the  child  tlien 
add  the  0  to  tlie  5,  aad  the  ansW'jr  will  be  1 1  cents. 
Eleven  cents  are  1  dime  and  1  cent.  Let  him  leave  1 
cent  under  the  column  of  cents,  and  substitute  1  dime  for 
the  10  cents.  Let  him  place  this  dime  with  the  2  dimes, 
and  his  answer  will  be  3  dimes  1  cent.  Ask  how  many 
cents  in  3  dimes  1  cent,  and  the  answer  will  be  31  cents. 
Thus  his  answer  will  be  either  3  dimes  1  cent,  or  31  cents. 
If  the  pupil  thus  sees  the  principle  once  illustrated,  by 


68  ARITHMETIC.       SECOND   PART. 

a  visible  process,  the  method  will  be  much  more  readily 
understood  and  remembered.  Let  the  following  sum  also, 
be  done  by  the  coins. 

Add  il,36  to  $2,97. 

Add  2$.  6d.  8  cts.  to  3$.  8d.  9  cts. 

Add  7  E.  2$.  5d.  6  cts.  to  4  E.  8$.  6d,  4  cts. 

Add  5d.  6  cts.  7m.  to  8d.  4  cts.  9m. 

Add  4  E.  0$.  6d.  5  cts.  to  5  E.  0$.  4d.  6  cts. 

Let  the  teacher  dictate  such  simple  sums  until  the  pro- 
cess of  writing  and  adding  is  well  understood,  and  can  be 
clone  with  rapidity  and  accuracy. 

Note  to  teachers. 

It  is  very  desirable  thatpupilsshould.be  required  to 
write  their  figures  with  accuracy  and  neatness,  and  learn 
to  place  them  in  strait  Imes,  both  perpendicular  and  hori- 
zontal. Also  that  they  learn  to  add  by  calculation,  and 
not  by  counting,  as  young  scholars  are  very  apt  to  do.  If 
a  teacher  will  but  he  thorough,  at  the  commencement,  in 
these  respects,  much  time  and  labor  will  be  saved. 

Mary  has  4  apples,  James  5,  and  Henry  7,  how  many 
have  all  together  ? 

One  boy  has  6  marbles,  another  4,  and  another  9,  how 
many  have  all  together  ? 

A  man  gave  9  cents  to  one  boy,  8  to  another,  and  11 
to  another,  how  many  did  he  give  to  all  ? 

10  and  11  and  9  are  how  many? 

12  and  7  and  4  are  how  many  ? 

4  and  5  and  7  are  how  many  ? 

One  man  owns  6  horses,  another  8,  and  another  9,  how 
many  have  they  all  ? 

In  a  school,  10  study  history,  11  geography,  and  15 
grammar,  how  many  scholars  in  the  whole  ? 

One  house  has  10  windows,  another  7,  and  another  12, 
how  many  are  tliere  in  all  ? 

James  lent  one  boy  8  cents,  another  6,  and  another  17, 
how  many  did  he  lend  them  all  ? 

If  a  lady  pays  7  dollars  for  a  veil,  9  dollars  for  a  dress, 
and  3  dollars  for  a  necklace,  what  amount  does  she 
«pend  ? 

6  and  9  and  18  are  how  many  1 


SIMPLE    ADDITION. 


69 


1 0  and  5  and  7  are  how  many  ? 
8  and  11  and  14  are  how  many? 
Let  the  pupil  be  taught  to  add  using  the  signs. 
the  hist  sum.     8+11  +  14  =^  33 


Thus 


Rule  fok  Simple  Addition. 

Flace  units  of  the  same  order  in  the  same  column,  ami  draw 
a  line  under.  Add  each  column  separately,  beffinnino-  at  the 
right  hand.  Place  the  units  of  the  amount,  tinder  the  column 
to  which  they  belong,  and  carry  the  tens  to  the  next  higher  or- 
der. 

Add  2694  and  3259  and  6438. 

Placing  units  of  the  same  order  in  the  same  column, 
they  stand  thus. 

2694 
3259 
6438 


12391 


Let  the  pupil  at  first  learn  to  add  ia  this  manner.  8 
units  added  to  9,  are  17,  and  4  are  21  units,  which  is  1  of 
the  unit  order,  to  be  written  under  that  order,  and  2  of  the 
order  of  tens,  to  be  carried  to  that  order.  2  tens  carried 
to  3  tens,  are  5,  and  5  are  10,  and  9  are  19  tens ;  which 
is  9  of  the  order  of  tens,  to  be  written  under  that  order, 
and  1  of  the  order  of  hundreds,  to  be  carried  to  that  order. 
Thus  through  all  the  orders. 

Add  the  following  numbers. 


0) 

22321 

(2) 
23432 

(3) 
110331 

(4) 
222311 

41332 

42212 

224212 

131232 

12123 

13124 

103123 

101221 

13220 

21101 
99869 

220320 

234031 

88996 

657986 

688795 

275496 

(6) 
456789 

(7) 
369543 

(8) 
4976432 

8732 

654321 

695432 

4976432 

70 


ARITHMETIC. 

FIRST  PART. 

54976 

456789 

567897 

6325498 

843215 

654321 

432591 

5192346 

7621 

543219 

526387 

8763945 

49673 

345678 

489549 

763497 

1239713 

3111117 

3081399 

30998150 

(9) 

(10) 

(11) 

(12) 

30648 

30430 

764325 

29367 

46469 

25895 

70504 

29367 

74057 

57644 

98469 

29367 

63396 

72919 

57157 

29367 

55275 

3647 

46946 

29367 

90534 

57246 

3284 

29367 

8953 
30142 

363 

247781 

176202 

1041048 

399474 

Let  the  pupil  now  learn  to  place  units  of  the  same  order 
in  the  same  column,  by  the  following  examples. 

Let  the  teacher  dictate  the  following.  The  pupils 
should  be  required  freviously  to .  attempt  writing  them, 
while  studying  their  lesson. 

1 

One  million,  four  hundred  and  sixty  thousand,  and  two. 

Twenty,  four  million,  six  hundred  and  one. 

Three  hundred  and  sixty  thousand,  four  hundred  and 
six. 

Ninety-four  million,  five  hundred  and  seventy-eight 
thousand,  three  hundred  and  forty-one. 

Six  million,  seven  thousand,  and  forty-three. 
2 

Two  hundred  and  six  thousand,  five  hundred  and  forty- 
two. 

One  million,  one  thousand,  and  one. 

Nine  hundred  and  ninety  million,  nine  hundred  and 
ninety-nine. 

Eighty-eight  thousand,   eight  hundred  and  eighty-eight. 

Ninety-nine  million,  seven  hundred  and  sixty-five  thou- 
sand. 


SIMPLE    ADDITION.  71 

3. 

Two  hundred  and  six  million,  five  thousand,  four  hun- 
dred  and  one. 

Fifty-six  million,  four  hundred  thousand,  five  hundred 
and  six. 

Three  billion,  ninety-nine  thousand,  and  four. 

Five  hundred  million,  thirty  thousand,  four  hundred  and 
forty. 

Seven   million,  six   hundred    and   fifty-four   thousand, 
three  hundred  and  seventeen. 
4. 

Four  million,  four  hundred  and  thirty-two  thousand,  one 
hundred  and  seventy-six. 

Forty-nine  thousand,  and  three. 

Nineteen  million,  seven  hundred  and  sixty-five  thou, 
sand,  nine  hundred  and  eighty-four. 

Five  hundred  and  ninety-one. 

Seven  hundred  and  sixty-three  thousand,  nine  hundred 
and  forty-three. 

Ninety-nine  million,  nine  thousand  and  ninety. 

5. 

Four  hundred  and  four. 

Five  million,  six  hundred  and  forty-three  thousand,  two 
hundred  and  seventeen. 

One  million,  and  two. 

Nine  thousand,  and  ninety-nine. 

Four  million,  five  hundred  and  seventy-six  thousand, 
three  hundred  and  eighty-four. 

Forty-four  million,  three  hundred  and  twenty-one  thou- 
sand, seven  hundred  and  four. 

6. 

One  hundred  million,  one  thousand,  and  ten. 

Nine  billion,  eight  hundred  thousand,  nine  hundred  and 
forty. 

Four  hundred  and  eighty-eight  million,  nine  hundred 
and  five  thousand. 

Eighty-eight  million,  seven  hundred  and  seventy-seven 
thousand,  and  nine. 

Nine  hundred  and  ninety-nine. 


72  ARITHMETIC.       SECOND  PART. 

Ninety-nine  million,  eight,  thousand,  and  four. 

Five  hundred  and  eighty-seven   million,  six  hundred 
and  forty-nine  thousand. 

Twenty-eight  thousand,  eight  hundred  and  ninety-nine. 

Four  hundred  thousand,  eight  hundred  and  seven. 

One  billion,  fifty-nine  million,  four  thousand  and  eighty- 
seven. 

8. 

Seven  hundred  million,  ninety-nine  thousand,  and  sev- 
enty-nine. 

Fifty-five  thousand,  seven  hundred  and  forty-four. 

Nine  million,  eight  hundred  thousand,  eight  hundred. 

Eight  thousand,  eight  hundred. 

Seven  billion,  and  seventeen. 
9. 

Eighty. four  thousand,  and  nineteen. 

Nine  miUion,  fifty-four  thousand,  seven  hundred. 

Seven  hundred  and  sixty-eight  thousand,  eight  hundred 
and  four. 

Four  billion,  twenty  million,  ten  thousand  and  fifly. 

Sixty  million,  two  hundred  thousand. 

Eleven  hundred  and  forty-two. 
10. 

Forty  thousand,  and  twelve. 

Nine  billion,  eight  thousand. 

Sixty  million,  seven  hundred  thousand,  and  ten. 

Nine   billion,  ninety  million,  eighty  thousand,  and  sev- 
enty-eight. 

Sixty-five  million,  and  four  hundred. 

One  billion,  and  four. 

11. 

Nine  hundred  thousand. 
Four  million,  fifty-five  thousand,  and  eighty. 
Three  hundred  and  sixty-four  thousand,  seven  hundred 
and  thirty-eight. 

Forty  million,  four  hundred  and  four. 
Six  hundred  and  thirty  thousand. 

12. 
Ten  million,  four  hundred. 


DECIMAL    ADDITION. 


73 


Seventy-six  thousand,  three  hundred  and  twenty-one. 
Eight   milHon,  forty-two  thousand,  six  hundred    and 
seventy-three. 

One  thousand,  four  hundred. 

Sixty-four  thousand,  three  hundred  and  twenty. 

One  bilHon,  and  seventy-three. 


DECIMAL  ADDITION. 
Rule  for  aui>i:ng  decimals. 
Place  figures  of  the  same  order  under  each  other.     Add 
each  column,  as  in  Simple  Addition,  and  in  the  answer  place 
a  separatrix  between  the  orders  oj  units  and  tenths. 
Example. 
What  is  the  sum  of  234,406.     4,6490.     13,234.  2,2. 
3650,4002.     99'J,4699. 

Placing  units  of  the  same  order  under  each  other,  they 
stand  thus  : — 

234,406 
4,6490 
13,234 
2,2 
3650,4002 
999,4699 


4904,3591 

Let  the  pu[)ils  proceed  a^  in  Simple  Addition,  caUing 
the  names  of  each  order,  thus  : — 

9  tens  of  thousandtiis  added  to  2,  are  11  tens  of 
tliousaadths  ;  which  is  1  ten  of  thous/wdlhs,  to  be  written 
under  that  order  ;  and  1  of  the  order  oi  thousandlhs,  to  be 
carried  to  that  order. 

1  thousandth  carried  to  9,  is  10,  and  4  are  14,  and  9 
are  23,  and  6  are  29  thousandths  ;  which  is  9  thousandths, 
to  be  written  under  that  order,  and  2  hundredths,  to  be 
carried  to  the  next  order. 

Thus  throufjh  the  other  orders,  observing  to  place  a 
separatrix  between  the  orders  of  units  and  tenths. 

Arrange  the  following    mixed  decimals  according  to 
their  orders,  and  then  add  them. 
7 


74  ARITHMETIC.       SECOND  PART. 

(1) 

306,42001.  20,3391.  3246,42.  .39,4695.  634,001. 
84,6302, 

(2) 
99,987.   65432,02564.   64,65.   596,32.   87632,- 
51739.  36,50.  51639,2154. 

(3) 
63,204.  6359,42591.  8642,39.  86423,2915.  68,241. 

(4) 
63,9876.  59432,1103.  95,02.  876,3254.  6634,251. 
3426,549. 

Let  the  pupil  write  and  add  the  following  sums  in  De- 
cimals. 

1. 

Four  units,  six  tenths,  four  hundredths,  five  thou- 
sandths. 

Two  tens,  four  units,  .six  hundredths. 

Three  tens,  two  units,  two  hundredths,  seven  thou- 
sandths. 

Six  units,  five  tenths,  seven  hundredths,  four  thou- 
sandths, three  tens  of  thousandfjis. 

One  unit,  three  tenths. 

2. 

Forty-two  units,  sixteen  thousandths. 

Five  units,  sixty-three  hundreds  of  thousandths. 

Seventy. four  units,  seven  thousand  five  hundred  and 
fifty-three  tens  of  thousandths. 

Two  units,  five  hundred  and  sixty  tens  of  thousandths. 
3. 

Two  hundred  and  forty-three  units,  two  hundred  and 
forty-three  thousandths,  seventeen  units,  nine  hundred 
and  seventy-three  tens  of  tliousandths. 

Fifty  units,  six  thousand  seven  hundred  and  forty-three 
hundreds  of  thousandths. 

Five  units,  eight  thousandths. 

One  thousand  units  ;  one  thousand  tens  of  thousandths. 
4. 

One  thousand  and  one  units  ;  one  thousand  and  one 
hundreds  of  thousandths. 


DECIMAL    ADDITION. 


75 


Nino  hundred  and   ninety-nine   units,   nine  thousand 
nine  hundred  and  thirty  hundreds  of  thousandths. 

Four  units,  thirty  tens  of  thousandths. 

Five  units,  fifty-five  thousand  and  forty-three  millionths. 
5. 

Sixteen  units,  seven  hundred  and  sixty-four  thousandths. 

Two  units,  forty-five  hundreds  of  thousandths. 

Fifty  units,  forty-two  miUionths. 

Seven  units,  nine  hundred  and  ninety -eight  tens  of  thou- 
sands. 

Six  units,  five  hundred  and  forty-nine  millionths. 
6. 

Four  thousand  units,  four  thousand  thousandths. 

Forty-one  units,  four  thousand,  four  hundred  and  nine 
hundreds  of  thousandths.  'v 

Seven  units,  eighty-seven  tens  of  thousandths. 

Four  hundred  and  forty-one   units,  ninety-nine  hun- 
dredths. 

Four  units,  four  hundreds  of  thousandths. 
7. 

Seventeen  units,  nine  thousand  eight  hundred  and  sixty 
hundreds  of  thousandths. 

Nine  units,  sixteen  tens  of  thousandths. 

Four  units,  fifty-five  hundredths. 

Sixty-three  units,  ninety-nine  millionths. 

One  unit,  seventy-four  thousandths. 
8. 

Five  hundred  and  forty-four  units,  eight  thousand  seven 
hundred  and  fifty-five  millionths. 

Ninety-nine    units,   four   hundred   hundreds    of   thou- 
sandths. 

Six  units,  eight  hundred  and  eighty-eight  thousandths.   . 

Eight  thousand  units,  seventy -four  tens  of  thousandths. 

Six  units,  eighty-eight  hundredths. 
9. 

Seventeen  units,  forty  thousandths. 

Five  units,  ninety-three  millionths. 

Forty-four  units,  eighty-seven  hundredths. 

Six  units,  nine  hundred  and  ninety-nine  thousandths. 

Four   hundred  and   twelve  units,  seventy -five  tens  of 
thousandths. 


76  ARITHMETIC.       SECOND   PART. 

10. 

Seventy-eight  units,  four  thousand  and  five  tens  of  thou- 
sandths. 

Two  units,  five  hundred  hundreds  of  thousandths. 

Seven  units,  eighty-nine  miilionths. 

Five  hundred  and  seventy-two  units,  seventy-six  thou- 
sand, eight  hundred  and  sixty-four  hundreds  of  thou- 
sandths. 

Nine  thousand  and  fifty  units,  nine  thousand  and  fifty 
miilionths. 

11. 

Five  hundred  and  eighty-seven  units,  twenty-nine  hun- 
dred tens  of  thousandths. 

Forty  units,  five  hundred  and  sixteen  miilionths. 

Eight  units,  four  hundred  and  ninety-six  thousand  miil- 
ionths. 

Five  hundred  and  forty-two  units,  two  thousand  hun- 
dreds of  thousandths. 

Seventeen  units,  nine  thousand  nine  hundred  hundreds 
of  thousandths. 

12. 

Sixty-five  units,  sixty-five  hundreds  of  thousandths. 

One  hundred  and  eighty  units,  one  hundred  and  eighty 
tens  of  thousandths. 

Twenty. four  units,  twenty -four  miilionths. 

Sixteen  units,  sixteen  hundredths. 

Five  units,  five  thousandths. 

Fifty  units,  fifty  hundreds  of  thousandths. 
13. 

One  hundred  and  seventy-six  units,  one  hundred  and 
seventy-six  hundreds  of  thousandths. 

Four  units,  two  thousand  four  hundred  and  seventy-five 
tens  of  thousandths. 

Eighty-four  units,  seven  hundred  and  sixty-three  mii- 
lionths. 

Two  hundred  units,  two  thousand  and  forty  tens  of 
thousandths. 

Seventeen  units,  four  thousand  and  four  miilionths. 
14. 

Seventy-four  units,  nine  hundred  and  eighty  miilionths. 

Four  units,  four  hundreds  of  thousandths. 


DECIMAL    ADDITION.  77 

Eighty-one  units,  nine  thousand  four  hundred  hundreds 
<of  thousandths. 

One  unit,  ninety  thousand  and  one  milHonths. 

Eleven  units,  one  hundred  tens  of  thousandths. 
15. 

Seventy  units,  seventy  thousandths. 

Five  units,  four  hundred  and  forty  hundreds  of  thou- 
sandths. 

Four  hundred  units,   seven  thousand  and  forty-three 
milHonths. 

Nineteen  units,  eighty  thousand  and  nine  milHonths. 

Six  units,  one    hundred  and   one  hundr/eds   of  thou- 
sandths. 

16. 

Nine  tenths,  four  hundredths,  three  tens  of  thousandths. 

Five  tens,  sixteen  thousandths,  four  milHonths. 

Forty  units,  one  hundredth,  ten  tens  of  thousandths. 

Seven  units,  five  tens  of  thousandtlis,  three  miliionths. 

Six  units,  four  tenths,  two  hundredths. 
17. 

Two  teas,  two  units,  nine  tens  of  thousandths. 

One  unit,  four  tenths,  two  hundredths,  seven  milHonths. 

Eight  tens,  two  hundredths,  six  hundreds  of  thousandths. 

Four  hundreds,  fourteen  miliionths. 

Six  units,  forty  thousand  hundreds  of  thousandths. 

Fifty-nine  units,  fifty-nine  thousand  miliionths. 
18. 

Eighteen   units,  four   hundred  and    sixtj'-three    thou- 
sandths. 

Nine  units,  eight  hundred  and  forty-three  miliionths. 

Twentv-two   units,   eleven  thousand  and  one  hundreds 
of  thousandths. 

Nine  units,  ninety-nine  hundreds  of  thousandths. 

Eighty-eight  units,  nine  milHonths. 

Four  units,  eight  hundred  and  eighty-eight  thousandths. 


METHODS   OF    PROVING    ADDITION. 

I..  Commence  at  the  top  instead  of  the  bottom  of  the  sev- 


/b  ARITHMETIC.       SECOND    PART. 

eral  columns,  and  if  the  same  answer  is  obtained,  it  may 
be  considered  as  right. 

2.  Draw  a  line  />nd  cut  off  the  upper  figure  of  all  the 
orders.  Add  the  lenainder  which  is  not  cut  off.  Then 
add  the  sum  of  this  rei>iainder  to  the  figures  cut  off,  and 
if  the  answer  is  the  same  as  the  first  answer,  it  may  be 
considered  as  right. 


COMPOUND  ADDITION. 

In  order  to  understand  the  following  sums,  the  pupil 
must  commit  to  memory  the  tables  inserted  in  the  com- 
mencement of  the  book. 

Sums  for  Mental  Exercise. 

If  a  man  has  2  lbs.  IC  oz.  of  beef,  and  buys  6  lbs.  8 
oz.  more,  how  much  ha§  he  in  the  whole  ? 

The  answer  will  be  8  lbs.  18  oz.  In  18  oz.  how  many 
pounds,  and  how  many  ounces  over  1  Set  down  the  oun- 
ces  that  are  over,  and  add  the  number  of  lbs.  to  the  8  lbs. 
and  what  is  the  answer  ! 

A  boy  has  3  yards  2  quarters  of  cloth,  and  buys  2  yards 
and  .3  quarters  more,  how  much  has  he  in  the  whole  ? 

One  man  buys  3  bushels  and  2  pecks  of  grain,  another 
buys  2  bushels  and  3  pecks,  how  much  do  both  together 
buy  ? 

If  you  have  1  quart  and  1  pint  of  milk,  and  buy  2  quarts 
and  i  pint  more,  how  much  will  you  have  ? 

One  rope  is  ?>  feet,  7  inches ;  another  is  4  feet,  6  inch- 
es ;  how  many  feet  are  there  in  both  together  ? 

If  2  weeks  4  days,  be  added  to  Iweek  5  days,  how  ma- 
ny weeks  will  there  be  in  all  ? 

If  6  pounds  9  oz.  be  added  to  5  pounds  8  oz.  how  ma- 
ny powuls  will  there  be  in  all  ? 

If  8  bushels  2  pecks,  be  added  to  4  bushels  3  pecks, 
iiow  many  bushels  will  there  be  ? 

If  7  yards  2  quarters,  be  added  to  8  yards  8  quarters, 
how  aiany  yards  will  there  be  ? 


COMPOUND   ADDITION.  79 

RUI.B    FOR    COMPOJND    ADDITION. 

Place  units  of  the  same  order  in  tfie  same  column.  Find 
the  sum  of  each  order.  Find  how  m<  y  units  of  the  ne.rt 
higher  order  are  contained,  in  the  svxi,  ^nd  carn^y  them  to  that 
order.    Set  the  reinainder  under  th'^ order  added. 

EXAMPLE. 
£.  S.  d. 
5  „  6  „  8 
4  „  9  „  9 
9  ,,  9  ,,  5 


19„5„10 

Let  the  pupil  add  thus  :  5  pence  added  to  9  are  14,  and 
8  are  22  pence.  This  sum  contains  1  of  the  order  of  shil- 
ling .^o  be  carried  to  that  order,  and  10  to  be  written  un- 
der iTie  order  added.  One  shilling  carried  to  9  makes 
10,  and  9  are  19,  and  6  are  25  shillings.  This  sum  con- 
tains  1  of  the  order  of  pounds,  to  be  carried  to  that  order, 
and  5  of  the  order  of  shillings,  to  be  written  under  that  or- 
der. 1  pound  carried  to  9  makes  10,  and  4  are  14,  and  5 
are  19  pounds,  which  are  written  under  that  order. 

Accustom  the  pupils  to  add  in  this  manner  ;  also  require 
them  to  separate  their  orders  in  Compound  Addition  by 
double  commas,  as  in  the  above  sum. 

Add  the  following  sums  : 

STKRLING    MONEY. 

£,.       s.       d.  £.       s.       d. 

12  „13  „  10  17  „  13  „  11 

14  „  9  „  9  13  „  10  „  2 

IG  „  6  „  .5  10  „  17  „  3 

18  „  12,,  11  8  „  8„  7 


TROY  WEIGHT. 

lbs.     oz.     jmt.  oz.     pwt.  gr. 

16  „  11  „  19  11  „  19  „  21 

4  „  4  „  16  10  „  16  „  8 

8„  8„  19  8„17„21 

6„  9„  14  6„  8  „23 


80  ARITHMETIC.       SECOND  PART. 

AVOIRDUPOISK    WEIGHSr, 

cwt.  qr.  lb.  lb.     oz.     dr. 

2„3v*27  24  „  13  „  14 

1  „1„1^  17„12„11 

4„2„26  26  „  12 „  15 

6„  1„13  16  „    8.,    7 


APOTHECARIES  WEIGHT. 

3.     9.    gr.  3.     3.     B. 

9  „  1  „  17  10  „  7  „  2 

3  „  2  „    9  6  „  3  „  0 
6  „  1  „  22  7  „  6  „  1 

4  „  0  „  16  9  »  5  „  2 


CLOTH    MEASURE. 

yd.     qr.  na.  E.  E.  qr.  na. 

71  „  3  „  3  44  „  3  „  2 

13„  2  „!•  49„4„3 

16  „  0  „  1  06  „  2  „  3 

42  „  3  „  3  84  „  4  „  1 


DRY  MEASURE. 

ph.    qu.  pt.  bu.     ph.  qt. 

1  „  7  „  1  17  „  2  „  5 

2  „  6  „  0  34  „  2  „  7 

1  „  5  „  0  13  „  3  „  6 

2  „  4  „  1  16  „  3  „  4 


WIXE    MEASURE. 

gal.     qt.  pt.  hhd.  gal.  ql 

39  „  3  „  1  42  „  61  „  3 

17  „  2  „  1  27  „  39  „  2 

24  „  8  „  0  9  „  14  „  0 

19  „  0  ,,  0  16  „  24  „  I 


COMPOUND    ADDITION.  81 

LONG    MEASURE. 

yds.  Jt.    in.  m.  fur.   po. 

4  „  2  „  11  46  „  4  „  16 

3  „  1  „    8  58  „  5  „  23 

1  „  2  „    9  9  „  6  „  34 

6  „  2  „  10  17  „  4  „  18 


LAND,    OR    SQUARE  JUlASUBE. 

acres,  roods,  rods.  ^I-Jf-   *?•  ^"' 

478  „  3  „  31  13  „  1446 

816  „  2  „  17  16  „  1726 

49  „  1  „  27  3  „  866 

63  „  3  „  34  14  „  284 


SOLID 

MEASURE. 

ton.    fL 

cords,  ft. 

41  „  43 

3  „  122 

12  „  43 

4  ,,114 

49  „  6 

7„  83 

4„27 

10  „  127 

TIME 

•■ 

y-     m.    to. 

h.     tnin.  sec. 

•'>'?  ,,  J  1  ;,  3 

23  „  54  „  32 

3„  9„2 

12  „  40  „  24 

29  „  8„2 

14  „  00  „  17 

46  „  10  „  2 

8„16„13 

CIRCULAR,  j 

MOTION. 

*•    o    . 

o     '     « 

3  „  29  „  17 

29  „  59  „  59 

1  »  6  „  10 

00  „  40  „  10 

4  „  18  „  17 

4  „  10  „  49 

6  „  14  „  18 

11  »  6„  10 

82  ARITHMETIC.       SECOND  PART. 

ADDITION  OF  VULGAR  FRACTIONS. 

Sums  for  Mental  Exercise. 

If  one  boy  has  one  half  an  orange,  and  another  three 
halves,  and  another  four  halves,  how  many  halves  are 
there  in  all  ? 

If  one  third  of  a  dollar,  five  thirds,  and  six  thirds,  be  ad- 
ded together,  how  many  are  there  in  all  ? 

One  man  owns  four  twentieths  of  a  building,  another 
six  twentieths,  and  another  eight  twentieths,  how  many 
twentieths  do  all  own  ? 

Seven  thirtieths,  nine  thirtieths,  and  six  thirtieths,  are 
how  many  ? 

Eight  twenty-fifths,  four  twenty-fifths,  and  seven  twen- 
ty-fifths, are  how  many  ? 


RULE    FOR  ADDING  VULGAR    FRACTIONS,     WHEN    ALL   HAVE 
THE    SAME    OR    A    COMMON    DENOMINATOR. 

Add  the  numerators,  and  place  their  sum  over  the  com- 
mon denominator. 

EXAMPLE. 

Add  ^3_  ^6_  _3_  and  ^V .  _    .        ,       ^ 
The  sum  of  the  numerators  is  15,  which  being  placed 
over  the  con>mon  denominator,  gives  the  answer  i|. 
Add  the  following  sums,  using  the  signs,  thus  : 

Arid    2       4     and    9  Ans     -2-   X  -*-   X  -®-   =J^. 

Add  j\  j\  and  If.         Add  ^\  j%  ,\  and  ^. 
Add  /j  ^\  and  -i^.         Add  |  f  f. 
When  fractions  having  a  different  denominator,  are  ad- 
ded,  it  is  necessary  to  perform  a  process  which  will  be  ex- 
plained  hereafter. 

Those  fractions  which  have  the  numerator  larger  than 
the  denominator,    are    called  improper  fractions,   thus : 

LP    X. 

When  we  use  the  expression  seven  halves,  we  do  not 
mean  seven  halves  of  one  thing,  because  nothing  has  more 
than  two  halves.  But  if  we  have  seven  apples,  and  take  a 
half  from  each  one,  we  shall  have  seven  halves ;  and  they 
are  halves  of  seven  things,  and  must  be  written  as  above. 


SIMPLE    SUBTllACTION.  83 

SUBTRACTION. 

There  are  four  kinds  of  Subtraction. 

The  Jirst  is  Simple  Subtraction,  in  which  the  minuend 
and  subtrahend  are  M'hole  numbers,  and  ten  units  of  one 
order,  make  one  unit  of  the  next  higher  order. 

The  second  is  Decimal  Subtraction,  in  which  the  minu- 
end  and  subtrahend  are  Decimals. 

The  third  is  Compound  Subtraction,  in  which  other  num- 
bers beside  ten,  make  units  of  a  higher  order. 

The  fourth  is  Subtraction  oj  Vulgar  Fractions,  in  which 
the  minuend  and  subtrahend  are  vulgar  fractions. 


SIMPLE  SUBTRACTION. 

If  8  cents  are  taken  from  12  cents,  what  will  remain  ? 

If  9  apples  are  taken  from  14  apples,  how  many  will  re- 
main? 

If  12  guineas  are  taken  from  20  guineas,  how  many 
will  remain  ? 

If  from  18  books,  12  be  taken,  how  many  will  remain  ? 

Let  the  following  examples  be  illustrated  by  the  coin  of 
the  U.  S. 

If  $2,  5d.  fi  cts.  be  taken  from  $3,  6d.  7  cts.,  how  much 
will  remain  ?  Which  is  the  subtrahend,  and  which  the 
minuend  ? 

Place  $3,  6d.  7  cts.  on  a  table,  side  by  side,  and  let  the 
pupil  take  the  amount  of  the  subtrahend  from  them. 

Subtract  $3,  4d.  5  cts.  from  $G,  7d.  7  cts. 

Subtract  3d.  4  cts.  2  m.  from  5d.  6  cts.  8  m. 

Subtract  8d.  7  cts.  5  m.  from  9d.  9  cts.  9  m. 

Let  the  teacher  place  on  the  table  the  coins,  thus  : 
$3,  4d.  C^  cts. 

Under  this  place  for  the  subtrahend,  the  following,  so 
that  the  coins  shall  stand  under  others  of  the  same  order.* 
$2,  2d.  4  cts. 

What  is  the  remainder,  when  the  value  expressed  by 
the  subtrahend,  is  taken  from  the  minuend  ? 

Now  if  10  cents  be  added  to  the  6  cents  of  the  min- 
uend, and  1  dime  be  added  to  the  2  dimes  of  the  subtra- 


*  The  pupil  must  understand  that  the  subtrahend  shows  how  many  of 
the  san^e  kinds  of  coin,  are  to  be  taken  from  the  minuend. 


84  ARITHMETIC.       SECOND  PART. 

hend,  will  there  be  any  difference  in  the  answer.  Let 
the  pupil  try  it  and  ascertain. 

If  10  dimes  be  added  to  the  4  dimes  of  the  minuend, 
and  1  dollar  be  added  to  the  2  dollars  of  the  subtrahend, 
will  there  be  any  difference  in  the  answer  ? 

Let  this  process  be  continued  until  every  member  of  the 
class  fully  understands  it,  and  then  let  them  commit  to 
memory  this  principle. 

"  If  an  equal  amount  he  added  to  the  Minuend  and  the 
Subtrahend  the  Remainder  is  unaltered. 

Let  the  following  coins  be  placed  as  minuend  and  sub- 
trahend. 


1 

d. 

cts. 

2 

1 

3  Minuend. 

1 

4 

5  Subtrahend, 

Which  is  the  largest  sum  taken  as  a  whole,  the  minuend 
or  subtrahend  ? 

If  each  order  is  taken  separately,  in  which  orders  is  the 
minuend  the  largest,  and  in  which  the  smallest  ? 
Can  you  take  5  cents  from  3  cents  1 
If  you  add  10  cents  to  the  3  cents,  you  c&n  subtract  5 
from  it,  but  what  must  be  done  to  prevent  the  Remainder 
from  being  altered  ? 

$     d.     cts.     m. 
From       4     3        2      4 
Subtract  14        5      6 

In  which  orders  are  the  numbers  of  the  subtrahend 
larger  than  those  of  the  minuend  ? 

Can  6  mills  be  taken  from  4  mills  ? 

What  can  you  do  in  this  case  ? 

If  10  mills  be  added  to  the  4  mills  of  the  minuend,  why 
must  1  cent  be  added  to  the  5  cents  of  the  subtrahend  ? 

From  6432,  subtract  3256, 

Can  6  units  be  taken  from  2  units  ? 

What  must  be  done  in  this  case  ? 


simple  subtraction.  85 

Rule  for  Simple  Subtraction. 
Write  the  subtrahend  under  the  minuend,  placing  units  of 
the  same  order  under  each  other,  and  draw  a  line  under. 
Subtract  each  order  of  the  subtrahend,  from  the  same  order 
of  the  minuend,,  and  set  the  remainder  under.  If  any  order 
of  the  subtrahend  is  greater  than  that  of  the  minuend,  add 
ten  units  to  the  minuend,  and  one  unit  to  the  next  higher  or- 
der of  the  subtrahend.     Then  proceed  as  before. 

•EXAMPLE. 

Subtract       4356 
From  2187 


21G9 


Let  the  pupil  subtract  thus  : 

Seven  units  cannot  be  taken  from  6  ;  therefore  add  10 
to  the  minuend,  which  makes  16.  7  from  16  leaves  9. 
As  10  units  have  been  added  to  the  minuend,  the  same 
amount  must  be  added  to  the  subtrahend.  1  of  the  order 
of  tens  is  the  same  amount  as  10  units,  we  therefore  add 
1  to  8  tens,  making  it  9  tens.  We  cannot  subtract  9  tens 
from  5  tens,  we  therefore  add  10  to  the  minuend,  which 
makes  15.  9  tens  from  15  leaves  6  tens.  As  10  tens 
have  been  added  to  the  minuend,  the  same  amount  must 
be  added  to  the  subtrahend — 1  of  the  order  of  hundreds  is 
the  same  amount  as  10  tens  ;  we  therefore  add  1  to  1 
hundred,  which  makes  2  hundred.  This  subtracted  from 
3  hundred  leaves  1  hundred. 

Thus  through  all  the  orders. 

Mode  of  Proof. 

A  sum  in  Subtraction  is  proved  to  be  right,  by  adding 
the  remainder  to  the  subtrahend ;  and  if  the  sum  is  the 
same  as  the  minuend,  the  answer  may  be  considered  as 
right. 

Let  the  following  sums  be  explained  as  above. 
Subtract 


34695 

from 

56943 

653215 

(( 

956432 

500032 

(( 

867200 

6291540 

« 

8732418 

354965 

a 

5360025 

8 

86  ARITHMETIC.       SECOND  FART. 

«  7985430  «  989763 

"  3542685  "  6542169 

"  5321543  "  7954324 

"  1223345  «  8500642 

"  1549768  «  3895463 

3543257  "  6385241 

"  2006935  "  5000623 

The  pupil  should  learn  to  subtract  by  the  use  of  the 
signs,  thus  : 

Subtract  5  from  7.         Ans.  7 — 5=2. 

Subtract  8  from  11.       Ans.  11 — 8=3. 

Subtract  the  following  numbers  in  the  same  way.  8 
from  17.  9  from  14.  6  from  20.  40  from  85.  800 
from  950.  1000  from  2744.  85  from  760.  95  from  700. 
440  from  763. 


DECIMAL  SUBTRACTION. 

If  2  tenths,  4  hundredths  of  a  dollar,  be  taken  from  4 
tenths,  6  hundredths,  what  will  remain  ? 

If  3  hundredths,  5  thousandths  of  a  dollar,  be  taken 
from  5  hundredths,  7  thousandths,  what  will  remain  ? 

If  5  dimes,  6  mills,  be  taken  from  7  dimes,  8  mills,  how 
much  will  remain  1 

If  4  dimes,  5  cents,  be  taken  from  7  dimes,  9  cents, 
how  much  will  remain  ? 

If  4  units,  6  tenths,  be  taken  from  6  units,  8  tenths,  how 
much  will  remain  ? 

In  simple  subtraction,  if  the  number  in  any  order  of  the 
minuend,  was  smaller  than  the  one  to  be  subtracted,  what 
did  you  do  ? 

The  same  is  to  be  done  in  Decimal  Subtraction. 

Take  4  tenths,  7  hundredths  of  a  dollar,  from  6  tenths, 
5  hundredths. 

In  which  order  is  the  number  of  the  subtrahend  the  lar- 
gest ? 

Can  7  hundredths  be  taken  from  5  hundredths  1  What 
must  be  done  in  this  case  ? 

Take  5  dimes,  6  cents,  from  8  dimes,  9  cents. 


DECIMAL    SUBTRACTION.  87 

In  which  order  is  the  number  of  the  subtrahend  the  lar- 
gest ? 

Can  9  cents  be  taken  from  6  cents  ?  What  must  you 
do  in  order  to  subtract  ? 

Subtract  7  hundredths,  8  thousandths  of  a  dollar,  from 
8  hundredths,  7  thousandths. 

Can  8  thousandths  be  subtracted  from  7  thousandths  ? 
What  must  be  done  in  this  case  1 


Rule  for  Decimal  Subtraction. 
Proceed  by  the  rule  for  common  Subtraction,  and  in  the 
answer  place  a  separatrix  between  the  orders  of  units  and 
tenths. 

EXAMPLE. 

Subtract   2,56  from  24,329.     Placing  the  subtrahend 
under  the  minuend,  so  that  units  of  the  same  order  stand 
in  the  same  column.     They  stand  thus  : 
24,329 
2,56 


21,769 

Let  the  pupil  learn  to  subtract  in  this  manner : 
Nothing  from  9  thousandths,  and  9  remains  to  be  set 
down.  6  hundredths  cannot  be  taken  from  2  hundredths  ; 
we  therefore  add  10  to  the  minuend,  which  makes  12.  6 
taken  from  12  leaves  6.  As  10  was  added  to  the  minu- 
end, an  equal  quantity  must  be  added  to  the  subtrahend. 

1  of  the  order  of  tenths  is  the  same  as  10  hundredths,  we 
therefore  add  1  to  the  5  tenths,  making  it  6  tenths.  6 
tenths  cannot  be  taken  from  3  tenths,  we  therefore  add 
10  to  the  minuend,  which  makes  13.  6  taken  from  13, 
leaves  7.  As  10  was  added  to  the  minuend,  an  equal 
amount  must  be  added  to  the  subtrahend.  1  of  the  order 
of  units  is  the  same  as  10  tenths,  we  therefore  add  1  to  the 

2  units,  making  it  8  units. 

Proceed  thus  through  all  the  orders,  remembering  to 
place  a  separatrix  between  the  orders  of  units  and  tenths. 

Let  the  following  sums  be  arranged  and  subtracted  in 
the  same  way  : 


88  ARITHMETIC.       SECOND  PART. 

Subtract  25,25  from     62,904 

"  790,4  "     96,409 

«        2,4693  «    354,268 

«         5,34689  "   ■   40,62 

«<         6,6543  "     23,3291 

«  432,54916  «    542,65329 

"  53,00300  «•    646,01201 

«  832,2  «  9988,659 

««  51,895  "     64,59432 

«         8,4156  "    400,21 

«  321,01013  "  4333,0063 

«  659,09543  "    679,2941 

1. 

Subtract  two  tens,  four  units,  three  tenths,  five  hun- 
dredths, and  four  thousandths  ;  from  four  tens,  two  tenths, 
five  hundredths,  and  four  thousandths. 

2. 

Subtract  two  tens,   three  units,   six  tenths,    nine  hun- 
dredths, and  three  thousandths,  from  four  tens,  four  units, 
three  thousandths,  and  five  tens  of  thousandths. 
3. 

Subtract  two  units,  four  thousand  three  hundred  and 
seventy.four  tens  of  thousandths  ;  from 

Twenty-three  units,  seven  thousand  five  hundred  tern 
of  thousandths. 

4. 

Subtract  ninety-eight  units,  two  thousand  nine  hundred 
and  eighty-seven  tens  of  thousandths ;  from 

Seven  hundred  and  seventy-seven  units,  four  thousand 
three  hundred  and  twenty-six  tens  of  iJwusandths. 

5. 

Subtract  seven  units,  six  thousand  five  hundred  and  for- 
ty-three tens  of  thousandths  ;  from 

Three  hundred  and  sixty-nine  units,  forty-two  hun- 
dredths. 

6. 

Subtract  seventy-seven  units,  twenty-four  tens  of  thou- 
sandths  ,•  from 


DECIMAL   SUBTRACTION.  89 

Two  hundred  and  twenty-five  uriits,  seven  thousand  six 
hundred  and  fifty-four  tens  of  thousandths. 

7. 

Subtract  twelve  units,  one  millionth ;  from 
Thirty  units,  ten  thousandths. 

8. 
Subtract  one  hundred  units,  eleven  tens  of  thousandths ; 
from 

Three  hundred  units,  one  tenth. 

9. 

Subtract  five  hundred  and  fifty  millionths ;  from 
Ninety-five  hundredths. 

10. 
Subtract  ninety-eight  units,  fifty-four  thousand  tens  of 
thousandths ;  from 

Eight  hundred  and  eighty-seven  units,  thirty-four  thou- 
sand tens  of  thousandths. 

11. 

Subtract  twenty  units,   seven   thousand  three  hundred 
and  twenty-one  tens  of  thousandths  ;  from 

Thirty-nine  units,  eighty-four  thousand,  three  hundred 
and  twenty-one  hundreds  of  thousandllis. 
12. 
Subtract  forty  units,  twenty-five  thousand,   nine  hun- 
dred and  eighty-three  hundreds  of  thousandths  ;  from 

Eight  hundred  and  forty-one  units,  six  hundred  and  for- 
ty.three  tens  of  thousandths. 

13. 
Subtract  eight  units,  forty-one  tens  of  thousandths ;  from 
Seventy,  seven  units,   forty -three  thousand  and  eleven 
millix>ntJis. 

14. 
Subtract  eight  units,  one  thousand  and  fourteen  mil. 
lionths ;  from 

Eight  hundred  units,  twenty -one  tensqfthousanths» 
8* 


90  ARITHMETIC.      SECOND   PART. 

15. 

Subtract  four  hundred  units,  sixty  hundredths ;  from 
One  thousand  units,  three  tenths. 

16. 
Subtract  fifteen  hundred  millionths ;  from 
Eighteen  hundreds  of  thousandths. 

17. 
Subtract  eighty  units,  eighty  thousandths ;  from 
Eight  hundred  units,  and  eighty  millionths. 

18. 

Subtract  two  units,  seventy-six  thousand  and  eight  mil- 
lionths ;  from 

Nine  hundred  and  eighty-seven  units,  forty-four  hun- 
dreds  oftliousandths. 


COMPOUND  SUBTRACTION. 

A  man  has  5  yds.  3  quarters  of  cloth,  and  cuts  off  2 
yds.  1  qr.  how  much  is  left  ? 

A  man  has  6  lbs.  3  oz,  of  beef,  and  sells  4  lb.  2  oz. 
how  much  is  left  ? 

If  4  bushels,  3  pecks,  are  taken  from  8  bushels,  5 
pecks,  how  many  remain  ? 

A  man  has  12  bushels,  6  pecks  of  grain  and  sells  7 
bushels  5  pecks,  how  many  will  remain  ? 

If  4  yards,  3  quarters,  2  nails,  be  taken  from  6  yds.  4 
qrs.  3  nails,  how  many  will  remain  ? 

If  4<£  „  3*.  „  Ad.  be  subtracted  from  6j£  „  8*.  „  bd. 
how  many  will  remain  ? 

If  the  same  quantity  be  added  to  the  minuend  and  sub- 
trahend, is  the  remainder  altered  ? 

Can  you  add  a  certain  quantity  to  the  minuend  in  one 
order,  and  the  same  quantity  to  the  subtrahend  in  another 
order  ?     Give  an  example. 

If  you  wish  to  subtract  1  yd.  3  quarters,  from  5  yds. 
2  qrs.  can  you  subtract  the  3  qrs.  from  the  2  qrs.  ? 

What  can  you  do  to  get  the  right  answer  ? 


COMPOUND    SUBTRACTION.  91 

If  4  shillings  4  pence,  be  taken  from  6  shillings  3 
pence,  how  many  will  remain  ? 

In  which  order  is  the  subtrahend  larger  than  the  minu- 
end ?  Can  4  pence  be  taken  from  3  pence  ?  What  must 
you  do  in  order  to  subtract  ? 

From  10  lbs.  8  oz.  subtract  9  lbs.  9  oz. 

In  which  order  is  the  subtrahend  larger  than  the  minu- 
end ?     What  must  be  done  in  this  case  ? 

From  7  feet  4  inches,  subtract  5  feet  6  inches. 

In  which  order  is  the  subtrahend  larger  than  the  minu- 
end ?     What  must  be  done  in  this  case  ? 


Rule  for  Compound  Subtractiow. 

Write  the  subtrahend  under  the  minuend,  placing  units  of 
the  same  order  under  each  other.  Subtract  each  order  of 
the  subtrahend,  from  the  same  order  of  the  minuend,  and 
set  the  remainder  under.  If  in  any  order  the  subtrahend  is 
larger  than  the  minuend,  add  as  many  units  to  the  minuend 
as  make  one  of  the  next  higher  order ;  then  add  one  unit  to 
the  next  higher  order  of  the  subtrahend. 

Example. 

Subtract  29£  19«.  Sd.  from  36£  15*.  Id. 
Placinu;  them  accordinor  to  rule  thev  stand  thus. 


£.         s. 
36     „   15 
29    „  19 

d. 

,,    8 

6     "   15 

»  11 

Subtract  thus  :  8  shillings  cannot  be  taken  from  7  ; 
therefore  add  as  many  units  of  this  order  to  7,  as  are  re- 
quired to  make  one  unit  of  the  next  higher  order  ;  that  is 
12  (as  12  pence  make  1  shilling).  12  added  to  7  are  19. 
Subtract  8  from  19,  and  11  remain  to  be  set  down. 

As  12  pence  have  been  added  to  the  minuend,  an  equal 
quantity  must  be  added  to  the  subtrahend  ;  therefore  car- 
ry 1  shilling  to  the  19  which  makes  20.  This  cannot  be 
subtracted  from  15  ;  therefore  ciJd  to  the  15  as  many  of 


02  ARITHMETIC.       SECOND  PART. 

this  order,  as  are  required  to  make  one  unit  of  the  next 
higher  order  ;  that  is  20.  This  being  added  to  15  makes 
35.  Subtract  20  from  35,  and  15  remain  to  be  set  down  ; 
as  20  shilHngs  have  been  addeti  to  the  minuend,  1  pound 
must  be  carried  to  the  subtrahend  of  the  next  higher  or- 
der,  which  makes  it  30  ;  and  this  subtracted  from  36, 
leaves  6  to  be  written  under  that  order. 

Let  the  following  sums  be  explained  as  above. 


Sterling  Monet. 

£. 
44 
36 

s.       d.                   s.       d. 
„  10   „  2                   16  „    8  „ 
„  11    „  8                   10  „    7  „ 

Trov  Weight. 

2 
4 

lb. 
6    „ 
2    „ 

oz.      pwt.                 oz.       pwt. 

11    „  14                    4    „    19 

3    „  16                    2    „    14 

Avoirdupois  Weight. 

„    21 
„    23 

e. 

7  „ 
5  „ 

qr.       lb.                   lb.       oz. 
3  „    13                    8    „    9    „ 
1  „    15                    6   „  12    „ 

Apothecaries  Weight. 

dr. 

12 

9 

3 

4  „ 
1   „ 

B       qr.                   3        3 
1  „    17                    10  „    3  „ 
,    2  „    15                      7  „   6  „ 

Cloth  Measure. 

3 
1 

1 

yd. 
35  „ 
10,. 

qr.       na.                  E.  E.       qr. 
1    „     2                       67    „     3 
1    „     3                        21    „     3 

Dry  Measure. 

na. 
„    1 

„   2 

hu. 
65  „ 
14  „ 

pk.       qt.                     pk.       ql. 
I    „    7                         2  „    3  „ 
3    „    4                        1  „    6  „ 

pt. 

0 

1 

SUBTRACTION. 

Wine  Measure. 

gal. 

qt. 

pt. 

hhd. 

^a^- 

?<■ 

21 

}i 

2 

>> 

a 

13 

>> 

0 

») 

1 

14 

>i 

2 

>> 

1 

10 

»> 

60 

» 

3 

Long  Measure. 

yd. 

J<- 

in. 

m. 

>r. 

!">• 

4 

»j 

2 

>> 

11 

41 

>> 

6 

»> 

22 

2 

» 

2 

» 

11 

10 

>> 

6 

» 

23 

93 


Land  or  Square  Measure. 

j4.     roods,     rods.  A.       r.      po. 

29    „    1    „     10  29  „  2  „  17 

24    „    1    „     25  17  „  1  „  36 

Solid  Measure. 

tons.      ft.  cords,      ft. 

116  „  24  72    „    114 

109  „  39  41    „    120 

Time. 

yrs.       mo.       we.  h.       min.       sec. 

54    „    11  „    3  20  „   41    „    20 

43    „    11  „    3  17  „    49   „    19 

CiRcuxAR  Motion. 


o 


9   „  23  „  45  29  „  34  „  54 

3  „     7  „  40  19  „  40  „  36 


SUBTRACTION  OF  VULGAR  FRACTIONS. 

If  a  boy  has  6  ninths  of  an  apple,  and  gives  away  4 
ninths,  how  much  remains  ? 

If  he  has  8  ninths,  and  gives  away  5  ninths,  what  re. 
mains  1 


94  ARITHMETIC.       SECOND  PART. 

If  he  has  7  twelfths,  and  gives  away  4  twelfths,  what 
remains  ? 

In  doing  these  sums  let  the  pupil  tell  first  which  is  the 
minuend  and  which  the  subtrahend. 

A  man  has  9  twentieths  of  a  dollar  and  loses  5  twenti- 
eths, how  much  remains  1 

If  he  has  11  twentieths  and  loses  7  twentieths,  what  re- 
mains ? 

If  he  has  8  sixteenths,  and  loses  5  sixteenths,  what  re- 
mains ? 

Subtract  /_  from  j\.     Subtract  ^\  from  if. 


Rule  foe  Subtracting  Vulgar  Fractions. 

Subtract  the  numerator  of  the  subtrahend,  from  the  nu- 
merator of  the  minuend,  and  place  the  remainder  over  the 
common  denominator. 

Let  the  pupil  in  doing  the  sums,  use  the  signs  in  this 
way. 

Subtract  |  of  a  dollar  from  |. 

An«!       ^ s.  =  3 

■  Subtract  A  from  A-     Subtract  ##  from  ||. 


_6JL 
9  0  0 

C( 

2  0  I 

900^* 

(< 

2±0 
48  0 

(( 

32  0 
4  8  0* 

1  6 

(( 

I  9 

<( 

1  6 

(( 

sa 

80 

8  0- 

30 

30" 

_96_ 
5  0  0  0 

(C 

_1_2_0_. 
5  000* 

u 

4 
20 

(( 

11 

A  man  owns  f  of  a  pasture,  and  sells  f ,  how  much  re- 
mains his  own "? 

A  boy  has  ii  of  a  guinea,  and  gives  away  g\,  how 
much  has  he  left  ? 

j\  from  If,  are  how  many  1  l^  from  fi^  are  how 
many  ? 

If  from  II  are  how  many  ?     /^  from  if  ?     yV  ^^o"^  if  • 


SIMPLE  MULTIPLICATION.     , 

Multiplication  is  repeating  a  number,  as  often  as  there 
are  units  in  another  number. 

The  number  to  be  repeated,  is  called  the  multiplicand. 


SIMPLE   MULTIPLICATION.  95 

The  figure  expressing  the  number  of  times  the  multipli- 
cand is  to  be  repeated,  is  called  the  multiplier. 

The  answer  is  called  the  product,  because  it  is  the  sum 
produced  by  multiplication. 

The  multiplier  and  multiplicand  are  called  the  factors, 
from  the  Latin  word  factum,  (made,)  because  they  are  the 
numbers  by  which  the  product  is  made. 

There  are  four  processes  of  multiplication. 

The  first  is  Simple  Multiplication,  where  the  factors  are 
whole  numbers,  and  ten  units  of  one  order  make  one  unit 
of  the  next  higher  order. 

The  second  is  Decimal  Multiplication,  where  one,  or 
both  the  factors  are  decimals. 

The  third  is  Compound  Mtdtiplication,  where  the  multi- 
plicand consists  of  orders,  in  which  other  numbers  be- 
sides  ten,  make  units  of  a  higher  order. 

The  fourth  is  the  multiplication  of  vidgar  fractions, 
where  one,  or  both  the  factors,  are  vulgar  fractions. 

A  boy  gives  8  apples  to  each  of  7  companions,  how 
many  does  he  give  to  them  all  ? 

A  man  travels  7  miles  an  hour,  how  far  will  he  travel  in 
9  hours  ? 

If  one  pound  of  raisins  cost  11  cents,  how  much  will 
6  pounds  cost  ? 

One  boy  has  7  cents,  and  another  twelve  times  as  many, 
how  many  has  the  last  1 

At  six  cents  apiece,  how  much  will  9  lemons  cost  ? 

At  12  cents  a  dozen,  how  much  will  8  dozen  marbles 
cost  ? 

One  pound  of  sugar  costs  9  cents,  how  much  will  5 
pounds  cost  ?     8  pounds  ?     11  pounds  ?     12  pounds  ? 

Multiplication  has  been  defined  as  repeating,  or  taking 
one   number  as  often  as  there  are  units  in   aiiotlier  num- 
ber.    Let  this  process  be  illustrated  by  the  coins  ;  thus, 
$  d.         cts. 

2    "     4     "     3 
Let  the  multiplier  be  2. 

Nov/ the  pupil  is  to  take  3  cents,  as  often  as  there  are 
units  in  2,  and  give  the  answer.  Then  he  is  to  take  4 
dimes  as  often  as  there  are  units  in  2,  and  then  2  dollars 
in  like  manner. 


96  ARITHMETIC.       SECOND    PART. 

Let  the  following  sum  be"done  by  the  coins. 
$  d.         cts. 

2     "    4    «     4 

Multiplied  by  3 

When  the  pupil  has  taken  4  cents  three  times,  he  will 
have  12  cents.  Let  a  dime  be  substituted  for  ten  of  these 
cents,  to  be  caried  to  the  next  product,  and  there  remain 
two  cents,  to  be  placed  in  the  order  of  cents.  Then  let  4 
dimes  be  taken  3  times,  which  make  12,  and  the  one  dime 
of  the  other  product  is  added,  making  13  dimes.  Let  a 
dollar  be  substituted  for  ten  of  the  dimes,  and  carried  to 
the  next  product,  and  three  dimes  will  remain  to  be  placed 
in  the  oi'der  of  dimes.  Two  dollars  taken  three  times, 
will  make  6  dollars,  and  adding  the  one  dollar  of  the  other 
product,  the  amount  is  7  dollars,  to  be  placed  in  the  order 
of  dollars. 

The  pupil  should  practice  in  this  way  until  the  principle 
is  fully  understood. 

Rule  for  muxtiplying,  when  the  multiplicand  has 
several  orders,  and  the  multiplied  does  not  exceed 

TWELVE. 

Place  the  multiplier  below  the  multiplicand.  Beginning 
at  the  right,  multiply  each  order  of  the  multiplicand,  by  the 
multiplier.  Place  the  units  of  the  product,  under  the  order 
multiplied,  and  carry  the  tens  to  the  next  product.  Write 
the  whole  of  the  last  product. 

Let  the  pupils  at  first  be  exercised  thus  : — 
Example. 
249 
8 


1992 
Eight  times  9  units  are  72  units  ;  which  is  2  units  to 
be  written  under  that  order,  and  7  tens  to  be  carried  to 
the  next  product.  Eight  times  4  tens,  are  32  tens,  and 
the  7  tens  carried,  make  39  tens,  which  is  9  of  the  order 
of  tens,  to  be  written  under  that  order,  and  3  hundreds  to 
be  carried  to  the  next  product.  Eight  times  2  hundreds, 
are  16  hundreds,  and  the  3  hundreds  carried,  make  19 
hundreds,  which  are  written  down. 


simple  multiplication.  97 

Examples. 

Multiply      348  by  4.  Multiply      2469  by  6. 

"              7ii8  "  5.  "              6923  "  7. 

««           4693  "  6.  "              4593  "  8. 

2914  "  7.  "  12468  "  9. 

3463  "  8.  "  42469  «  10. 

"            6798  "  9.  "  532-;  3  "  5. 

5124  "  10.  "  65492  "  8. 

8763  "  11. 

When  the  7nulfiplier  consists  of  several  orders,  another 
method  is  adopted.     For  example^** 

Multiply  324  by  67.  • 

The  324  is  first  to  be  multiplied  by  the  7  units,  accord, 
ing  to  the  former  rule,  and  the  figures  stand  thus, 

324 
67 


2268 


The  324  is  now  to  be  multiplied  by  the  6  ;  what  is  the 
number  represented  by  the  6  ?     Ans.     60  or  6  tens. 

If  4  is  multiplied  by  6  tens,  the  answer  is  24  tens,  or 
240.  The  4  is  to  be  written  in  the  order  of  tens,  under 
the  6,  and  the  2  (which  is  200)  is  to  be  carried  to  the  next 
product.     See  below. 

324 
67 


2268 
1944 


21708     Ans. 

The  2  tens,  or  (20)  are  next  multiplied  by  the  6  tens, 
(or  60)  and  the  answer  is  12  hundreds,  (1200)  and  the  2 
hundreds  to  be  carried  to  it,  make  1400.  The  4  is  writ- 
ten in  that  order,  and  the  1  carried  to  the  next  product. 
Next  the  3  hundreds  are  multiplied  by  the  6  tens,  and  the 
answer  is  18  thousands,  (18000)  and  the  1  to  be  carried 
9 


98  ARITHMETIC.       SECOND  PART. 

to  it,  make  19  thousand,  which  are  placed  in  their  orders. 
Then  the  two  products  are  added  together,  and  the  an- 
swer is  obtained. 

Let  the  pupil  answer  the  following  questions  on  the 
above  sum. 

What  number  does  the  6  of  the  multiplier,  represent  ? 
What  number  does  the  2  represent  1  If  they  are  multi- 
plied together,  as  if  they  were  units,  what  i§the  product? 
How  many  ciphers  must  be  added,  to  express  the  true 
value  of  2  tens,  multiplied  by  6  tens  ?  Elow  many  fig- 
ures are  at  the  right  hand  of  both  the  factors,  2  tens  and  6 
tens  ?  Is  the  number  of  ciphers  added,  the  same  as  the 
number  oi  figures  at  the  right  hand  of  both  the  factors? 

What  is  the  answer  if  the  3  hundreds  be  multiplied  by 
6  tens,  as  if  they  were  units  ?  How  many  ciphers  must 
be  added,  to  make  the  product  express  the  true  value  ? 
Does  the  number  of  ciphers  added,  correspond  to  the 
number  of  figures,  at  the  right  of  both  factors  ? 

By  answering  the  above  questions,  the  pupil  will  un- 
derstand the  following  principle  ? 

Figures  of  any  order  may  he  multiplied  together  like 
units,  and  the  true  value  is  found,  by  annexing  as  many  ci- 
phers, as  there  are  figures  at  the  right  ofi  both  the  fac- 
tors. ^ 

Let  the  following  questions  be  answered. 
Multiplicand  f'OO 
Multiplier        237 

What  number  is  represented  by  6  ?  by  3  ? 

If  the  6  is  multiplied  by  the  3,  what  is  the  answer,  if 
the  factors  are  considered  as  units  ?  What  is  the  true  an- 
swer ? 

If  the  2  is  multiplied  by  3,  what  is  the  answer  if  they 
are  considered  as  units  ?  what  is  the  true  answer  ? 

What  number  is  represented  by  2  ?  by  8  ?  If  the  2  is 
multiplied  by  8,  what  is  the  answer  if  they  are  considered 
as  units  ?     What  is  the  true  answer  ? 

Let  the  pupil  now  learn  to  multiply  the  above  sum,  and 
place  the  figures  in  the  orders  to  which  they  belong  ; 
thus, 


V 


SIMPLE    MULTIPLICATION,  99 

869  Multiplicand. 
237  Multiplier. 


6083 

2607 
1738 

205952  Answer. 

The  multiplicand  is  first  multiplied  by  the  7  of  the  mul- 
tiplier, and  the  product  is  6083. 

Then  the  3  tens  (or  30)  are  multiplied  into  the  9  units, 
and  the  answer  is  270  ;  which  is  7  tens  to  be  set  in  the 
order  of  tens,  and  2  hundreds  to  be  carried  to  the  next 
product.*  Then  the  6  tens  (or  60)  are  multiplied  by  3 
tens,  and  the  product  is  1800,  and  the  2  that  were  to  be 
carried  make  2000  ;  which  is  2  of  the  order  of  thousands 
to  be  carried  to  the  next  product,  and  0  to  be  set  in  the  or- 
der  of  hundreds.  Then  the  8  hundreds  are  multiplied  by 
3  tens,  and  the  answer  is  24000,  and  the  2  to  be  carried 
make  26000  ;  which  is  6  to  be  set  in  the  order  of  thou- 
sands, and  2  in  the  order  of  tens  of  thousands. 
'  Next  take  the  2  hundred  as  multiplier,  and  multiply  9 
units  by  it,  and  the  answer  is  1800  ;  which  is  8  to  be  set 
in  the  order  of  hundreds,  and  1  to  be  carried  to  the  next 
product. 

Proceed  thus,  till  all  the  orders  have  been  multiplied  by 
the  2  hundred.  Then  add  the  several  products  and  the 
answer  is  obtained. 


Rule  for  Simple  Multiplication,  when  the  mul- 
tiplier HAS  SEVERAL  ORDERS. 

Place  the  multiplier  below  the  multiplicand,  so  that  units  of 
tlie  same  order,  may  stand  in  the  same  column.  Multiply 
by  each  order  of  the  multiplier.  Write  the  units  of  each 
product,  in  the  order  to  which  they  belong,  and  carry  the 
tens  to  the  next  product.  Add  tlie  products  of  the  several 
orders,  and  the  sum  is  the  answer. 

*  The  cipher  is  omitted,  because  as  the  figure  is  set  under  the  8, 
we  can  tell  what  order  it  belongs  to,  without  the  cipher. 


100  arithmetic.     second  pari". 

Example. 

826 
234 


3304 
2478 
1652 

193284 

Multiply  by  the  4  units  according  to  the  other  rule. 

Then  multiply  each  order  of  the  multiplicand  by  the  3 
tens  (or  30)  thus :  6  units  multiplied  by  3  tens  are  18  tens, 
which  is  8  tens  to  be  written  in  that  order,  and  1  of  the 
order  of  hundreds  to  be  carried  to  the  next  product.  2 
tens,  (or  20)  multiplied  by  3  tens  (or  30)  are  600,  and 
the  iOO  carried,  makes  700,  which  is  7  to  be  written  in 
the  order  of  hundreds.  8  hundreds  multiplied  by  3  tens, 
(or  30)  is  24000  ;  which  is  4,  to  be  written  in  the  order  of 
thousands,  and  2  tens  of  thousands  to  be  set  in  that  order. 

Lastly,  multiply  each  order  of  the  multiplicand  by  the  2 
hundreds.  6  units  multiplied  by  2  hundreds,  are  12  hun. 
dreds,  which  is  2  hundred  to  be  written  in  that  order,  and 
1  thousand  to  be  carried  to  the  next  product.  2  tens  (or 
20)  multiplied  by  2  hundreds,  are  4000,  and  the  1000  car. 
ried  makes  5000,  which  is  5  to  be  placed  in  the  order  of 
thousands.  8  hundreds  multiplied  by  2  hundreds,  are 
160,000,  which  is  6  tens  of  thousands,  to  be  written  in  that 
order,  and  1  hundred  of  thousands,  to  be  written  in  the 
order  of  hundreds  of  thousands. 

Add  all  the  orders  of  the  products,  by  the  rule  of  com- 
mon addition,  and  the  sum  is  the  answer. 


Multiply 


Examples 

256 

by  26    Ml 

3639 

"  329 

4638 

"  462 

5943 

''  567 

2345 

"  234 

7892 

"  456 

If  five  stands  alone  (5)  of  what  order  is  it  ? 


4567 

by 

234 

4654 

496 

6789 

596 

5432 

281 

45G8 

362 

8382 

945 

•  is  it  ? 

If 

a  ci- 

SIMPLE    MULTIPLICATION.  101 

tplier  is  affixed,  of  what  order  is  it  ?  How  much  larger  is 
the  sum,  than  it  was  before  ?  By  what  number  was  it 
multiphed  when  the  cipher  was  added  ? 

If  two  ciphers  are  added  to  the  5,  in  what  order  will  it 
stand  ?  Hf)w  much  larger  is  the  sum  than  it  was  before  ? 
By  what  numbei  was  it  multiplied  when  the  ciphers  were 
added  ? 

If  three  ciphers  are  added  to  5,  in  what  order  will  it 
stand  1  How  much  larger  is  the  sum  than  it  was  before  ? 
By  what  number  was  it  multiplied  when  the  ciphers  were 
added  ? 

If  you  wish  to  multiply  5,  by  10,  what  is  the  shortest 
way?  If  you  wish  to  multiply  5,  by  100,  what  is  the 
shortest  way  ?  If  you  wish  to  multiply  5,  by  1000,  what 
is  the  shortest  way  ? 

If  you  wish  to  multiply  50,  by  2,  how  would  you  do  it  ? 

Would  it  make  any  difference  if  you  should  multiply 
the  5  first,  and  then  affix  a  cipher  to  the  answer  1 

If  you  are  to  multiply  5000,  by  2,  can  you  begin  by  mul- 
tiplying the  5  first  ? 

If  you  are  to  multiply  35000,  by  2,  can  you  multiply 
the  5'first,  and  then  the  3,  and  afterwards  affix  the  three 
ciphers  ? 

If  you  are  to  multiply  20  by  30,  can  it  be  done  by  mul- 
tiplying the  3  and  2  together,  and  then  affixing  2  ciphers 
;to  the  product  ? 

Multiply  200  by  20  in  the  same  way. 


RuxK  roil  Multiplying  when  tuk  factors  are  tek- 

MINATED  BY  CirilKRS. 

Multiply  the  significant  figures  together,  and  to  their  jpro- 
■ducf  annex  as  many  ciphers  as  termhiate  both  the  Jactors^ 
Note. — All  figures  are  called  significant,  except  ciphers. 

Multiply 


30 

by  20 

400 

by 

00 

sooo 

9 

96 

i-i 

30 

200 

6 

4400 

.u 

.90 

2000 

"   40 

100 

'«  100 

• 

2400 

"  2000) 

160 

■«  4200) 

102  ARITHMETIC.       SECOND  PART. 

When  any  number  is  made  by  multiplying  two  numbers 
together,  it  is  called  a  composite  number. 

Thus  12  is  a  composite  nurabei',  because  it  is  made  by 
multiplying  3  and  4  together. 

Is  18  a  composite  number?  What  two  numbers  multi- 
plied together  make  18? 

Is  14  a  composite  number  ?  Is  13  a  composite  num- 
ber ?     Is  9  a  composite  number  ? 

If  12  is  multiplied  by  8,  what  is  the  product?  What 
are  the  factors  which  compose  8  ? 

If  you  multiply  12  by  one  of  these  numbers,  and  the 
product  by  the  other,  will  the  answer  be  the  same  as  if 
you  multiplied  12  by  8  ? 

Let  the  pupil  try  and  see. 

What  are  the  numbers  that  compose  18  ? 

Multiply  123  by  18.  Multiply  it  by  one  of  the  num- 
bers that  compose  18,  and  the  product  by  the  other  num- 
ber, and  what  is  the  result  ? 


Rule  for  multiplying,  when  the  multiplier  ex- 
ceeds 12,  ANU  IS  A  COMPOSITE  NUMBER. 

Resolve  the  multiplier  into  the  factors  which  compose  it, 
end  multiply  the  multiplicand  by  one,  and  the  product  by  the 
other. 

Let  the  following  sums  be  done  by  the  above  rule. 

Multiply         33     by     20        Multiply       587     by     16 


268   < 

'   49 

a 

6543 

a 

24 

329 

'   54 

a 

521 

a 

27 

426 

'   32 

C( 

72 

i( 

30 

2345 

'   96 

(( 

793 

a 

36 

7G54 
fi543 

"   64 
'  4ni 

In  multiplication  it  makes  no  difference  in  the  product, 
which  of  the  factors  is  used  for  multiplier  or  multiplicand  ; 
for  3  times  4,  and  4  times  3,  give  the  same  product,  and 
thus  with  all  other  factors.  It  is  in  most  cases  most  con- 
veaient  to  place  the  largest  number  as  multiplicand. 


-  DECIMAL    MULTIPLICATION.  103 

DECIMAL  MULTIPLICATION. 

In  explaining  decimal  multiplication,  it  is  needful  to  un- 
derstand the  mode  of  multiplying  and  dividing  by  the  «cp- 
aratrix. 

If  we  have  2,34  we  can  make  it  ten  times  greater,  by 
moving  the  separatrix  one  order  to  the  right,  thus,  23,4. 
For  23  units,  4  tenths,  is  ten  times  as  much  as  2  units,  34 
hundredths. 

It  is  therefore  multiplied  l)y  10. 

We  can  multiply  it  by  100  by  removing  the  separatrix 
entirely,  thus,  234,  for  the  2  units  and  34  hundredths,  be- 
come 234  units,  and  are  thus  multiplied  by  100. 

Whenever  therefore  we  wish  to  multiply  a  mixed  or 
pure  decimal,  by  any  number  composed  of  1  and  ciphers, 
we  can  do  it  by  moving  the  separatrix  as  many  orders  to 
die  right,  as  there  arc  ciphers  in  the  multiplier. 

EXAMPLES. 

Multiply     402,5946  by  100 

"  2,6395  "  1000 

«  4,03956  "  10000 

«'  54,0329  «  10 

«'  4,6930  «  1000 

"  3694  «  100 

«  4,6934  «  10000 

But  if  the  decimal  has  not  as  many  fgures  at  the  right, 
as  are  needful  in  moving  the  separatrix,  ciphers  can  be  ad- 
ded thus.  Multiply  2,5  by  1000.  Then  in  order  to  muU 
tipiv  by  a  cipher,  it  is  necessary  to  move  the  separatrix 
as  many  orders  to  the  right,  as  there  are  ciphers  in  the 
multiplier,  1000,  in  order  to  do  this,  two  ciphers  must  bo 
added  thus, 

2500, 

Here  2  U7iits,  and  5  tenths,  are  changed  to  2  thousands 
and  bhundreds,  and  of  course  are  made  1000  times  lar- 
ger, or  multiphed  by  1000. 

In  the  following  examples,  in  order  to  multiply  by  mo- 
King  the  separatrix,  it  is  necessary  to  add  ciphers  to  the 
right  of  the  multiplicand. 


104  ARITHMETIC.       SECOND   PART. 


EXAMPLKS. 


Multiply         3,7  by        100 
«  2,35    "       1000 

"  2,5   "     10000 

"      34,200   «  100000 


Multiply         5,2  by        100 

36,3  "       1000 

3,869  "     10000 

"      5,6469  "  100000 


Division  also,  can  be  performed  on  decimals,  by  the  use 
of  the  separatrix. 

Whenever  we  divide  a  number,  we  make  it  as  much 
smaller,  as  the  divisor  is  greater  than  one. 

If  we  divide  by  10,  as  10  is  ten  times  greater  than  one, 
we  make  the  number  10  times  smaller. 

If  we  divide  by  100,  we  make  the  numbers  100  times 
smaller. 

If  therefore  we  make  a  number  10  or  100  times  smaller, 
we  divide  by  10  or  100. 

If  we  make  it  1000  times  smaller,  we  divide  by  1000, 
&c. 

If  then  we  are  to  divide  323,4  by  10,  we  must  make  it 
10  times  smaller.  This  we  can  do  by  moving  the  sepa- 
ratrix  one  order  to  the  left,  thus,  32,34.  If  we  are  to  di- 
vide by  100,  we  can  do  it  by  moving  the  separatrix  two  or- 
ders  to  the  left,  thus,  3,234. 

If  we  lire  to  divide  by  10,000,  we  can  do  it  by  moving 
the  separatrix  4  orders  to  the  left,  thus,  ,3234. 

Whenever  therefore,  we  wish  to  divide  a  pure  or  mixed 
tiecimal,  by  a  number  composed  of  1  and  ciphers,  we  can 
do  it  by  moving  the  separatrix  as  many  orders  to  the  left^ 
as  there  are  ciphers  in  the  divisor. 


Divide 


But  if  the  decimal  has  not  enough  figures  to  enable  the 
separatrix  to  be  moved,  according  to  the  rule,  ciphers 
must  be  prefixed-^ 

Thus  jf  we  wish  to  divide  3,2  by  100,  we  do  it  thus, 
,082.  Here  the  3  is  changed  from  3  iniits,  to  3  hundredths. 
and  of  course  made  100  times  less. 


EXAMPLES. 

32,5  bv    10 

342,6  "    100 

469,3  "   1000 

46936,7  "  10000 

234()9,8  "  lOOOnO 

Divide   32,69  by     10 

3269,1  "    100 

2396,4  "    1000 

"   12346,95  "   10000 

"   15463,90  "  100000 

DECIMAL    MULTIPLICATION. 


105 


Divide 


.  EXAMPLES. 

2,4  by     100 

Divid« 

3  23,4  by 

10000 

32,4  ' 

1000 

24G,4  " 

100000 

932,5  ' 

10000 

293,6  " 

100000 

21,6  ' 

100000 

546,9  " 

100000 

600,7  ' 

1000000 

32,3  " 

1000000 

286,9  ' 

10000000 

100,4  " 

10000000 

542,8  ' 

100000000 

3094,9  " 

1000000 

A  decimal  can  also  be  multiplied,  by  expunging  the  sep- 
aratrix. 

Thus  2,4  is  multiplied  by  10,  by  expunging  the  separa- 
trix,  thus,  24, 

2,56  is  multiplied  by  100,  by  expunging  the  separatrix, 
thus,  256. 

In  all  these  cases,  the  decimal  is  multiplied  by  a  num- 
ber composed  of  1,  and  as  many  cquhers  as  there  are  deci- 
mals at  the  right  of  the  separatrix  Avhich  is  expunged. 

If  yow  expunge  the  separatrix  of  the  following  decimals, 
by  what  number  are  they  multiplied  ? 

2,40.  3,295.  54,6823. 

54,63.       89,46321.  5,l>432. 

How  can  you  multiply  3,1  by  10  ?  What  is  it  after 
this  multiplication  ? 

How  do  you  multiply  3,12  by  100  ?  What  is  it  after 
this  multiplication? 

How  do  you  multiply  9,567  by  1000  ?  What  is  it  after 
this  multiplication  ? 

If  the  separatrix  is  expunged  from  2,52,  by  what  is  it 
multiplied  ? 

If  the  separatrix  is  expunged  from  2,56934,  by  what  is 
it  multiplied  ? 

If  the  separatrix  is  removed  from  5,9432 1 6,  by  what  is 
it  multiplied  1 

If  the  separatrix  is  removed  from  3,46^ ,  by  what  is  it 
multiplied  ? 

If  the  separatrix  is  removed  from  3,5..  by  what  is  it 
multiplied  ? 

If  a  man  supposes  he  owes  $54,23,  and  finds  he  owes 
10  times  as  much,  what  is  the  sum  he  owes?  How  do 
you    perform  the    multiplication    with    the   separatrix  ? 


106  ARITHMETIC.       SECOND  PART. 

What  does  the  number  become  aftei;  being  thus  multipli- 
ed? 

Multiply  in  the  above  mode  $244,635  by  10,  by  100, 
and  by  1000.  What  does  the  sum  become,  by  each  of 
these  operations  ? 

Divide  ^244,635  by  10,  by  100,  and  by  1000,  with  the 
separatrix.  What  does  the  sum  become  by  each  of  these 
operations  ? 

Divide  and  multiply,  with  a  separatrix,  $25.')6,436,  by 
10,  by  100,  and  by  1000. 

If  before  multiplying,  the  multiplieand  is  made  a  certain 
number  of  times  larger,  the  product  is  made  as  much  lar- 
ger. If  the  multiplicand  is  made  too  large,  the  product  is 
as  much  too  large. 

For  example  ; 

If  we  wish  to  find  how  much  twice  2,3  is,  we  can  change 
it  to  whole  numbers,  and  multiply  it  by  2,  and  we  know 
the  answer  is  10  times  too  large.  For  23  is  10  times  lar- 
ger than  2,3,  and  therefore  when  it  is  multiplied  by  2,  its 
product  is  10  times  too  large.  If  then  we  make  it  10 
times  smaller,  we  shall  have  the  right  answer.  When- 
ever therefore,  we  wish  to  multiply  a  decimal,  we  can 
change  it  to  whole  numbers,  and  multiply  it  by  the  rule 
for  common  multiplication.  We  then  can  make  the  pro- 
duct as  much  smaller,  as  we  made  the  multiplicand  larger, 
by  changing  it  to  whole  numbers. 

For  instance,  if  we  wish  to  multiply  3,6  by  3,  we  can 
expunge  the  separatrix,  and  the  multiplicand  becomes  10 
times  too  large.  We  then  multiply  it  as  in  whole  nura- 
.bers  thus, 

36 
d 

108 

This  product  is  also  10  times  too  large,  and  we  find  the 
right  answer,  by  placing  a  separatrix  so  as  to  divide  it  by 
10,  thus  making  it  ten  times  smaller. 

In  like  manner,  if  the  multiplier  is  increased  a  certain 
number  of  times,  the  product  is  increased  in  the  same  pro- 
portion. 


DECIMAL   MULTIPLICATION. 


107 


If  we  are  to  multiply  32  by  2,3,  and  should  hy  expung- 
ing  the  separatrix,  change  the  multiplier  to  whole  num- 
bers, it  would  make  the  product  10  times  too  large,  and  to 
obtain  the  right  answer  we  must  divide  the  product  by  lO 
with  a  separatrix,  thus  making  it  10  times  smaller. 

Multiply  2,5  by  4. 

By  what  number  do  you  multiply,  when  you  expunge 
the  separatrix  of  the  decimal  ? 

What  is  the  product  of  the  multiplication  after  the  sep- 
aratrix is  expunged  ?  How  much  too  large  is  this  pro- 
duct? 

How  do  you  divide  this  product  by  the  same  number  as 
;ou  multiplied  the  decimal  ? 

Explain  each  process  as  above. 

Multiply  12,46  by  5     Multiply  3,2  by  6 

18,23  "  8  "  52,23  «  7 

«  ,346  "  9  "  286,45  «'  8 

"  36,2  "  7  "  123,678  "  9 

«  25,36  "  5  "  32,92  "  12 

«  44,429  "  4  "  64,64  "  11 

«  92,1234  "  7  "  988,931  «  9 

Multiply       329     by       2,4     Multiply       764    by    8,925 

72.63 

984,4 

6,529 

,462 

,3596 

Let  the  multiplier  be  2,  4,  and  the  multiplicand  is  3,6. 
Changing  the  multiplier  to  whole  numbers,  would  make 
the  product  ten  times  too  large.  Should  the  mulfiplicand 
be  changed  to  whole  numbers,  the  product  would  again 
be  made  ten  times  larger,  so  that  it  would  be  made  100 
times  too  large.  Therefore  to  bring  the  answer  right,  we 
must  divide  it  by  100,  thus  making  it  100  times  smaller. 
This  is  done  by  the  use  of  a  separatrix.  3,6  and  2,4, 
when  changed  to  whole  numbers  and  multiplied  together, 
are  864.  This  is  100  times  too  large,  and  is  brought  right, 
by  dividing  it  by  100  thus,  8,64. 


329 

by   2,4 

Ml 

jltiplv   764 

426 

3,5 

"  ■'   2875 

362 

"   39,5 

"   30021 

4689 

"   2,36 

8643 

4693 

"    5,462 

2875 

2678 

"  6,H246 

7628 

108  ARITHMETIC.       SECOND   PART. 

iRuLE  FOR  DECIMAL  MULTIPLICATION. 

Change  the  Decimals  to  whole  numbers  by  expunging  the 
separatrix.  Multiply  as  in  whole  numbers.  Divide  the  an- 
swer by  the  product  of  the  two  numbers  by  which  the  J  actors 
were  multiplied,  in  expunging  the  separatrix. 

EXAMPLE. 

Multiply  8,61  by  4,7. 

Change  these  to  whole  numbers,  and  they  become  801 
and  47.  (Here  the  multiplicand,  in  expunging  the  sepa- 
ratrix, is  multiplied  by  100,  and  the  multiplier  by  10.) 
Multiplying  them  together,  they  produce  40467.  The 
product  of  the  two  numbers  bv  which  the  factors  were 
multiplied,  (10  and  100),  is  1000.  Dividing  40467  by  it, 
gives  the  answer  40,407. 

EXAMPLES. 

Multiply  2,37  by  4,6. 

By  what  do  you  multiply  each  factor  when  you  remove 
the  separatrix  ?     What  is  the  product  of  the  two  numbers 
by  which  you  multiplied  the  factors  ? 
How  do  you  divide  by  this  product  ? 

Multiply  2,64     by  3,8 

"  362,68      "  48,72 

"  6895,40      "  3,651 

«  334,02      "  28,54 

«         2195,334     "  3,2 

«         3456,567      "  ,51 

«  937,8      "  ,84 

«         1234,636      "  36,4 

"  765,3      "  1,23 

89123,002      «  ,591 

The  following  common  rule  for  decimal  multiplication, 
includes  all  the  others,  and  may  be  used  after  understand- 
ing the  preceding. 


RULE    FOR    DECIMAL    MULTIPLICATION. 

Multiply  as  in  whole  numbers,  and  then  point  off  in  the 
product,  as  many  orders  of  decimals,  as  are  found  in  both 
the  factors. 


COMPOUND  MULTIPLICATION.  109 


Multiply 


EXAMPLES. 

3,69 

by    3,8 

Multiply 

,12 

by 

4,6 

8,600 

.'      5,9 

(( 

1,94 

a 

,600 

224,7 

"      2,3 

11 

351,9 

cc 

6 

9,427 

"      3,4 

cc 

,658 

<( 

,236 

COMPOUND  MULTIPLICATION. 

If  4  grains,  3  penny-weights,  are  repeated  3  times, 
what  is  the  product  ? 

If  3  yards,  1  quarter,  be  repeated  3  times,  what  is  the 
product  ? 

If  5  feet,  2  inches,  be  repeated  4  times,  what  is  the 
product  ? 

If  2  hogsheads,  5  gallons,  be  repeated  5  times,  what  is 
the  product? 

If  4  drams,  2  ounces,  be  repeated  3  times,  what  is  the 
product  ? 

What  is  4  times  2  days,  7  hours  ? 

What  is  5  times  3  months,  4  days  ? 


RULE    FOR   COMPOUND    MULTIPLICATION. 

Place  the  multiplier  below  the  multiplicand.     Multiply 
each  order  separately,  beginning  with  the  lowest.     In  the 
product  of  each  order,  find  hoiv  many  units  there  are  of  the 
next   higher  order.     Carry  these  tinits  to  the  next  product, 
and  setihe  remainder  under  the  order  multiplied. 
£.       s.     d. 
1   "    9  "  6 
4 


5  "  18  "0 
Proceed  thus: — Four  times  six  pence  are  24  pence, 
which  is  2  units  of  the  next  higher  order,  (or  shillings,) 
to  be  carried  to  that  order  ;  and  as  no  pence  remain,  a  ci- 
pher is  to  be  placed  in  the  order  of  pence.  Four  times 
9  shillings  are  36  shillings,  and  the  2  carried  make  38 
shillings,  which  is  1  pound,  to  be  carried  to  the  next  pro- 
10 


110  ARITHMETIC.       SECOND  PART. 

duct,  and  10  shillings  to  be  written  in  the  shilling  order. 
Four  times  1  pound  is  4  pounds,  and  the  1  carried,  makes 
5,  which  is  to  be  written  in  the  order  of  pounds. 

Let  the  pupil  do  the  following  sums,  stating  the  process 
while  doing  it,  as  above. 

What  cost  9  yards  of  cloth,  at  5s.  6d.  per  yard  ? 

What  cost  5  cwt.  of  raisins,  at  £1,  3s.  3d.  per  cwt.  ? 

What  cost  4  gallons  of  wine,  at  8s.  7d.  per  gallon  ? 

What  is  the  weight  of  6  chests  of  tea,  each  weighing  3 
cwt.  2  qrs.  9  lbs.  ? 

What  is  the  weight  of  7  hogsheads  of  sugar,  each 
weighing  9  cwt.  3  qis.  12  lbs.  ? 

How  much  brandy  in  9  casks,  each  containing  41  gals. 
3  qts.  1  pt.  ? 

yds. 

1.  Multiply   14 

hhd. 

2.  Multiply  21 

le. 

3.  Multiply  81 

a. 

4.  Multiply  41 

yr. 

5.  Multiply  20 

S. 

6.  Multiply  1 

yds. 

7.  Multiply     3 

1.  In  .35  pieces  of  cloth,  each  measuring  27 J  yds.  how 
many  yards?  Ans.  971yds.  1  qi\ 

2.  In  9  fields,  each  containing  14  acres,  1  rood,  and  25 
poles,  how  many  acres  ?  Ans.   129  a.  2  q)-s.  25  rods. 

3.  In  6  parcels  of  wood,  each  containing  5  cords  and 
96  feet,  how  many  cords  ?  Ans.  34  cords,  64  feet. 

4.  A  gentleman  is  possessed  of  li  dozen  of  silver 
spoons,  each  weighing  2  oz.  15  pwt.  11  grs.,  2  dozen  of 
tea-spoons,  each  weighing  10  pwt.  14  grs.,  and  2  silver 
tankards,  each  21  oz.  15  pwt.  Pray  what  is  the  weight  of 
the  whole  ?  Ans.  8  lb.   10  oz.  2pwi.  6  grs 


ANSWERS. 

.     qr.  na. 

[      3   2  by  11 

yds.     qr.  na. 
163   2  2 

!.  g.      qt.  pt. 
15   2  1  by  12 

lihd.  g.   qt.  pt. 
254  61  2  0 

m.  fur.  po. 

2   6   21  by  8 

le.  m,  fur.  po. 
655  1  4   8 

r.     p> 

2  11  by  18 

a.     r.     p. 
748  0  38 

m.  w.  d. 

yr.     m.  w.  d. 

5  3  6  by  14 

286  5  2  0 

0    /    II 

S.     "    '    '' 

15  48  24  by  5 

7  19  2  0 

r.  ft. 

87  by  8 

yds.    ft. 
29  56 

MULTIPLICATION    OP    VULGAR    FRACTIONS.  Ill 

MULTIPLICATION  OF  VULGAR  FRACTIONS. 

MULTIPLICATION    WHEN    ONLY    THE    MULTIPLICAND    IS    A 
FRACTION. 

A  man  gave  one  child  three  quarters  of  a  dollar,  and 
another  four  times  as  much,  how  much  did  he  give  the 
last? 

A  man  has  12  barrels  of  wine,  and  takes  a  half  pint 
from  each  3  times,  how  many  half  pints  does  he  take  ? 

If  a  man  has  an  ounce  of  silver,  and  takes  2  sixteenths 
from  it  6  times,  how  many  sixteenths  does  he  take  1 

How  much  is  4  times  two  sixths  ? 

How  much  is  5  times  two  sixths  ?     6  times  1     7  times  ? 

From  the  above  examples  it  appears,  that  we  can  mul- 
tiply a  fraction  by  a  whole  number,  by  multiplying  its  nu- 
merator. 

Let  the  pupil  perform  the  following  sums,  first  mental- 
ly, and  then  on  the  slate. 

1.  What  is  9  times  -^^  1 

2.  What  is  3  times  y^  ? 

3.  What  is  6  times  ^\  ? 

4.  What  is  7  times /^? 

5.  What  is  8  times  g5_  ? 
<3.  What  is  7  times  j\\  1 

7.  What  is  3  times  3^^  ? 

8.  What  is  5  times  -^^  ? 

9.  What  is  8  times  /^  ? 

10.  What  is  4  times  j\  ? 

11.  What  is  6  times  ^%  ? 

12.  What  is  5  times  -^^  ? 

13.  What  is  4  times  j\1 

14.  What  is  8  times  ^? 

15.  What  is  5  times  /^  ? 

16.  What  is  6  times  ^\  1 

17.  What  is4  times/_? 

18.  What  is  3  times  j%  ? 

19.  What  is  9  times  /g  ? 

20.  What  is  6  times  /^  ? 

In  performing  these  sums  on  the  slate,  let  the  pupil  use 
the  signs,  thus  : 


112  ARITHMETIC.       SECOND    PART. 

Two  twentieths  multiplied  by  nine,  equals  eighteen 
twentieths  ;  and  is  expressed  by  signs  as  follows : 

_2_     V    Q   18. 

2  0      ^^    ^   2  0 

There  is  another  method,  by  which  the  value  of  a  frac- 
tion is  multiplied,  by  increasing  the  size  of  the  parts  ex- 
pressed by  the  denominator. 

For  example,  when  we  wish  to  multiply  j\  by  2,  the 
most  common  way  is  to  multiply  the  numerator  by  2, 
thus  : 

12^'*'  1  2 

But  the  same  effect  is  produced,  if  we  divide  the  denom- 
inator by  2,  thus  : 

_4_  V  2  —  i 

1  2    A    ■*  6 

It  will  easily  be  seen,  that  y^^  and  |  are  the  sa7ne  quan- 
tity.  The  only  difference  is,  that  in  one  case  the  unit  is 
divided  into  12  parts  and  8  are  expressed,  and  in  the  oth- 
er case,  the  unit  is  divided  into  6  parts,  and  4  are  ex- 
pressed. In  one  case,  we  make  twice  as  many  pieces, 
and  in  the  other  we  make  them  twice  as  large. 

When  we  multiply  the  numerator,  the  number  nf  parts  is 
multiplied,  and  when  we  divide  the  denominator  the  size 
of  the  parts  is  multiplied. 

If  we  multiply  j\  by  3,  in  what  two  ways  can  it  be 
done  ? 

If  we  multiply  the  numerator,  what  is  it  that  is  multi- 
plied 1 

If  we  divide  the  denominator,  what  is  it  that  is  multi- 
plied  ? 

Multiply  f  by  3  in  both  ways,  and  tell  what  each 
method  multiplies. 


Rule  for  multiplying  when  only  the  multiplicand 

IS  A  fraction. 

Multiply  the  numerator,  or  divide  the  denominator  hy  the 
multiplier. 

Let  the  following  sums  be  performed,  and  explained  as 
above. 


MULTIPLICATION  OP  VULGAR  FRACTIONS.  113 


Multi 

ply  tV 

by 

4 

Multiply  1 

by 

2 

(< 

_4_ 
I  8 

9 

(< 

t\ 

(C 

7 

(( 

sV 

6 

« 

/o 

(( 

7 

(( 

^ 

9 

« 

eV 

<< 

8 

<( 

3^0 

10 

(( 

5 

(( 

3 

(( 

/l 

7 

u 

'6 

TO 

(( 

10 

(( 

_9_ 
4  0 

5 

(( 

/o 

« 

5 

<( 

4^ 

8 

(( 

A 

(( 

8 

<< 

4% 

6 

(( 

4=^8 

(( 

6 

(( 

_6_ 

9 

(( 

4 

.S   "i 

(C 

U 

I 


Multiplication  where  oaly  the  multiplier  is  a 
Fraction. 

1.  If  you  have  twelve  cents,  and  give  away  a  sixth  of 
them,  to  each  of  four  children,  how  many  cents  do  you 
give  away  ? 

Ans.  A  sixth  of  twelve  cents  is  two  cents.  Two  cents 
given  to  each  oijour  children  would  be  eight  cents  given 
away. 

2.  If  a  man  has  fifteen  cents,  and  gives  a  fifth  of  them, 
to  each  of  three  children,  how  many  does  he  give  away  ? 

Ans.  One  fifth  of  fifteen  is  three.  Three  times  three 
is  nine.     He  gives  away  nine  cents. 

From  the  above  examples  it  appears  that  when  we  mul- 
tiply by  a  fraction,  wo  take  a  part  of  the  multiplicand,  and 
repeat  it  a  certain  number  of  times.  In  the  last  case  the 
man  had  fifteen  cents,  which  is  the  multiplicand.  We 
take  a.  fifth  of  it  and  repeat  it  three  times. 

3.  If  a  man  had  eighteen  cents,  and  gave  a  ni7ith  of 
them,  to  six  difierent  boys,  how  many  cents  did  he  give 
away  ? 

In  the  above  question,  what  is  the  muUiplicand  ?  What 
part  are  you  to  take  from  it,  and  how  often  are  you  to  re- 
peat it  ? 

4.  If  a  man  has  twelve  dollars,  and  gave  a  fourth  of 
them  to  three  different  workmen,  how  many  did  he  give 
awav  ?  What  is  the  multiplicand  ?  What  part  are  you 
to  take  from  it,  and  how  often  are  you  to  repeat  it  ? 

5.  How  do  you  multiply  itvelve  by  three  fourths  ? 
Ans,  We  take  a  fourth  of  twelve  and  repeat  it  three 

10* 


114  ARITHMETIC.       SECOND  PART. 

times.     One  fourth  of  twelve  is  three.     Three  fourths  ate 
three  times  as  much.     Three  times  three  is  nine. 

6.  How  do  you  multiply  eighthy  three  fourths  ? 

7.  How  do  you  multiply  eighteen  by  three  ninths  ? 

8.  If  you  multiply  twelve  by  three,  do  you  make  it  larger 
or  smaller  1  if  you  multiply  it  by  three  fourths,  do  you 
make  it  larger  or  smaller  ? 

Why  is  the  multiplicand  made  smaller  when  you  multi- 
ply by  three  fourths  ? 

Ans.  Because  we  do  not  repeat  the  whole  number,  but 
only  a  fourth  part  of  it ;  and  this  is  repeated  only  three 
times,  which  does  not  make  it  as  large  a  number  as  the 
multiplicand. 

9.  If  you  multiply  eight  by  three,  do  you  make  it  larger 
or  smaller  ?  If  you  multiply  it  by  three  fourths,  do  you 
make  it  larger  or  smaller  ?     Why  ? 

10.  M\x\\\}^\y  fifteen  by  two  thirds. 

1 1.  Multiply  twenty  four  by  five  sixtlis. 

12.  Multiply  thirty-two  by  three  eighths. 

13.  MixXix^ly  fourteen  by  three  sevenths. 

14.  Multiply  sixteen  by  two  eighths. 

15.  Multiply  twenty  four  hy  five  sixtJis. 

Multiplication  has  been  defined,  as  repeating  a  number, 
as  often  as  there  are  units  in  another  number. 

In  multiplying  by  a  fraction,  we  take  such  a  part  of  a 
number,  as  is  expressed  by  the  denominator,  and  repeat  it 
as  often  as  there  are  units  in  the  numerator. 

Thus  in  multiplying  12  by  a  we  take  a  sixth  part  of  12, 
and  repeat  it  4  times,  and  the  answer  is  8, 

Note.— Tlie  propriety  of  calling  the  number  in  the 
numerator  units,  is  explained  on  page  40,  where  the  dis- 
tinction is  shown  between  units  that  are  whole  numbers, 
and  units  that  are  fractions.  It  is  shown  also  on  page  57, 
where  it  appears  that  the  numerator  ma}^  be  considered 
as  whole  numbers,  divided  by  the  denominator. 

In  multiplying  let  the  pupil  u.se  the  signs  thus  : — 

Multiply  12  by  |. 

12  -~  6  =  2 
2  X  3  =  6.     Answer. 


MULTIPLICATION  OP  VULGAR  FRACTIONS.  115 

In  doing  the  above  sum  what  part  of  12  is  taken  ?  How 
often  is  it  repeated  '.' 

Is  the  product  larger  or  smaller  than  the  multiplicand  ? 

Multiply  12  by  f . 

Is  I  a  proper  or  improper  fraction  ? 

Is  there  a  whole  unit  in  f  ? 

Is  the  product  larger  or  smaller  than  the  multiplicand, 
when  12  is  multiplied  by  f  ? 

Why  is  it  larger  when  multiplied  by  |  and  smaller  when 
multiplied  by  J  ? 

Lot  the  following  sums  be  stated  thus ;  16  X  f .  One 
fourth  of  16  is  4,  and  two  fourths,  is  twice  as  much,  or  8. 
Multiply         16     by     i     by     a     by     I 


16 

hy 

I    k 

18 

I     « 

a 

24 

1        (1 

8 

36 

9 

42 

1           << 

7 

C3 

1            (1 

y 

Examples  for  Mental  Exercise. 

If  you  have  14  apples,  and  give  one  seventh  of  them 
to  each  of  four  boys,  how  many  do  you  give  away  ? 

Ans.  A  seventh  to  1  boy,  would  be  2,  and  four 
sevenths,  would  be  four  times  as  much,  or  8. 

What  is  4  of  14  ? 

If  you  have  48  cents,  and  give  a  twelfth  of  them,  to 
«ach  of  two  boys,  how  many  do  you  give  awaiy  ? 

What  is  j%  of  48  ? 

A  man  has  35  sheep,  and  sells  four  fifths  of  them,  how 
many  does  he  sell  ? 

A  boy  has  40  marbles,  and  loses  |  of  them,  how  many 
does  he  lose  ? 

What  is  40  multiplied  by  f  ? 

What  is  36  multiplied  by  j%  ? 

What  is  f  of  21  ?  f  of  24  ?^  f  of  81  !  4  of  49  ?  f  of  641 
I  of  16  ?  4  of  40  ?  I  of  45  ?  -pVof  60  ?  /^  of  96  ?  |  of  24  ? 
i  of  30?  " 

What  is  ^  of  18  ?  ^\  of  100  ?  |  of  40  ?  f  of  28  ?  f  of 
27  ?  -r\  of  33  ?  f  of  4'8  ?  f  of  81  ?  j\  of  144  1    ||  of  99  t 

What  is  f  of  54?  ^  of  49  ?  f  of  32  ?  f  of  81  ?  j%  of 
70  ?  j\  of  88  ?  j?-  of  96  1  f  of  16  ?  I  of  12  ?  I  of  18  ?  j% 
©f24?  I  of  15?  ^3_of36? 


116  arithmetic.     second  part. 

Examples  for  the  slate. 


112^  X  A 
144  X  r% 

2608  X  U 
720  X  ^\ 

1335  X  II 
578  X  t\ 


1912  X  if 
1357  X  U 
545  X  M 
722  X  A 
304  X  If 
420  X  A 


If  a  number  is  to  be  both  multiplied  and  divided  by  two 
figures,  it  makes  no  difference  which  is  done  Jirst,  provi- 
ded  the  same  figures  are  used  as  multiplier  and  divisor. 

For  example,  let  it  be  multiplied  by  2,  and  dividedhy  9. 

We  can  divide  first  by  9,  and  then  multiply  the  quo- 
tient by  2  ;  or  we  can  multiply  first  by  2,  and  then  divide 
the  product  by  9,  and  the  answer  is  the  same. 

Thus  18  multiplied  by  2,  is  36  ;  and  this  divided  by  9 
is  4. 

Again  18  divided  by  9,  is  2  ;  and  this  multiplied  by  2 
is  4. 

If  then  we  multiply  12  by  |  we  divide  by  4,  to  find  one 
fourth  of  13,  and  multiply  by  3,  to  obtain  three  fourths, 
and  the  answer  is  9.  But  if  we  sliould  multiply  12  by  3, 
and  then  divide  the  product  by  4,  the  answer  would  be 
the  same.  Thus  12  x  3  =  3G  and  36  -r-  4  =^  9.  Thus 
9  is  the  same  answer  as  is  obtained  by  dividing  12  by  the 
denominator,  and  multiplying  the  answer,  by  the  numera- 
tor. What  are  the  two  ways  in  which  18  can  be  multi- 
plied by  f  ?  What  will  be  the  answer,  if  it  is  divided  by  6 
first,  and  the  quotient  multiplied  by  4  ?  What  will  be  the 
answer,  if  it  is  multiplied  by  4  first,  and  then  the  product 
divided  by  6  ? 


EuLE  FOR  MULTIPLYING  WHEN  ONLY  THE  MULTIPLIER 
IS   A   FRACTION. 

Divide  by  the  denominator,  to  obtain  one  part,  and  multi- 
ply by  thenvmerator,  to  ohiain  the  required  number  of  parts. 

But  in  case  this  division  should  leave  a  remainder; 

Multiply  by  the  numerator  Jirst,  and  tlicn  divide  the  pro- 
duct by  the  denominator. 


MULTIPLICATION  OF   VULGAR    FRACTIONS. 


117 


EXERCISES  FOR  THE  SLATE. 


In  all  these  cases  it  is  best  to  imtUiply  by  the  numerator 
Urst,  and  then  divide  by  the  denominator.  If  any  remains 
after  division,  place  the  divisor  under  it,  for  a  fraction. 


Multiply, 


13G9  by  f 

6463  "  I- 

3264  "  f 

43256  "  f 

86432  «  I 

3549  "  I 

54683  "  # 


Multiply 


4681  by  I 

3042  "  f 

5963  "  I 

46938  "  4 

G3921  "  -pV 

26438  "  f 

39621  «  # 


EXAMPLES  FOR  MENTAL  EXERCISE. 

1.  If  15  is  five  eighths  of  some  number,  what  part  of 
15  is  one  eighth  of  that  number? 

2.  If  12  is  four  sixths  of  some  number,  what  part  of 
12  is  one  sixth  of  that  number  ? 

3.  If  18  is  six  ninths  of  some  number,  what  part  of  18 
is  one  ninth  of  that  number? 

4.  If  15  is  I  of  a  number,  what  is  \  of  that  number  ? 

5.  If  15  is  I  of  some  number,  what  is  that  number  ? 

Let  such  exercises  he  stated  thus. 

6.  If  15  is  five  eighths,  a  fifth  of  15  is  one  eighth.  A 
fifth  of  1 5  is  3.  If  3  is  one  eighth  then  the  whole  is  8 
times  as  much,  or  24. 

7.  24  is  I  of  what  number  ? 
"8.  36  is  I  of  what  number  ? 

9.  42  is  ^  of  what  number  ? 

10.  If  a  man  can  do  |  of  a  piece  of  work  in  12  days, 
how  long  would  it  fake  him  to  do  |  of  it  ? 

(Ans.)  It  would  take  liim  only  one  sixth  of  the  time,  to 
do  one  seventh  that  it  does  to  do  f .  i  of  12  is  2.  It 
would  take  him  2  days. 

Let  the  remaining  sums  be  stated  as  above. 

11.  If  a  man  bought  ^  of  a  barrel  of  wine  for  18  dol- 
lars, how  much  will  J-  cost  ? 

12.  How  much  will  the  whole  cost? 


118  ARITHMETIC.       SECOND   PART. 

13.  Bought  I  of  a  chaldron  of  coal  for  24  shillings, 
how  much  will  i  cost  ?     How  much  will  the  whole  cost  ? 

14.  If  15  is  f  of  some  number,  what  is  one  eighth  of 
that  number  ? 

15.  "What  is  the  whole  of  that  number?  If  23  is  f  of 
one  number,  what  is  that  number  ? 

16.  If  a  man  bought  -f  of  a  cask  of  brandy  for  42  dol- 
lars, what  is  1  worth  ?  what  is  the  whole  worth  ? 

17.  If  I  of  a  month's  board  cost  12  dollars  what  is  it  a 
month  ? 

18.  If  I  of  a  cord  of  wood  cost  16  shillings,  what 
would  i  cost  and  what  would  the  whole  cost  ? 

19.  28  is  1  of  what  number  ? 

20.  48  is  I  of  what  number  ? 

21.  56  is  f  of  what  number  ? 

22.  32  is  y\  of  what  number  ? 

23.  How  many  times  is  4  contained  in  5  ? 
(Ans.)  Once  and  one  over. 

24.  What  is  i  of  1  ?  What  is  i  of  1  ? 

25.  What  is  i  of  1  ?  What  is  i  of  2  ?  What  is  |  of  2  ? 

26.  What  is  i  of  12  ?  What  is  i  of  1  ?  What  is  i  of  2? 
What  is  I  of  4  ?  What  is  |  of  4  ? 

27.  What  is  f  of  4?    What  is  f  of  2?    What  is  |  of  1? 

28.  How  many  units  in  |  of  2? 

29.  How  many  units  in  |  of  3  ? 

30.  How  many  units  in  |  of  5  ? 

31.  How  many  units  in  ^  of  11  ? 

32.  How  many  units  in  |  of  6? 

33.  How  many  units  in  |  of  18  ? 

34.  How  many  units  in  |  of  16  ? 

35.  How  many  units  in  '^  of  21  ? 

It  will  be  seen  that  in  fractions,  as  in  whole  nunlbers,  it 
makes  no  difTerence  in  the  product,  which  factor  is  used 
as  multiplier. 

For     12  X  I  =  9 
And    f    X  yV=  V  =  9 

Here  when  the  whole  number  is  used  as  multiplier,  the 
answer  is  an  improper  fraction,  which,  if  changed  to  whole 
numbers  is  9. 

Multiply  18  by  |  and  |  by  18,  and  tell  in  what  respects 
the  answers  differ 


MULTIPDICATION  OE    VULGAR   FRACTIONS.  119 

Multiply  ^  by  14,  and  14  by  ^. 

Is  there  any  difference  in  tlie  value  of  the  answers  ? 

In  what  respect  do  they  differ  ? 


Multiplication    where    both    factors    are    feac- 

TIONS. 

1.  If  we  had  i  an  orange  and  should  give  away  half  of 
this  i  what  part  of  an  orange  should  we  give  away  ? 

How  much  is  1  of  ^  ? 

2.  If  we  have  i  of  an  orange,  and  should  give  away 
i  of  it,  what  part  "of  a  whole  orange  should  we  give 
away  ? 

(Ans.)  If  the  two  halves  of  any  thing  be  divided  into 
4  pieces  each,  the  whole  is  divided  into  8  pieces.  Ta- 
king i  of  one  of  these  halves  then,  is  taking  i  of  the 
whole. 

3.  If  we  have  i  of  an  orange,  and  give  away  half  of  it, 
what  part  of  the  whole  orange  do  we  give  away  ? 

Ans.  If  an  orange  is  divided  into  4  pieces,  and  each  of 
these  pieces  are  haired,  the  orange  is  divided  into  8  pieces, 
and  each  piece  is  i  of  the  whole. 

i  of  1  is  i. 

4.  If  you  receive  i  of  an  orange,  and  you  give  |  01  it 
away,  what  part  of  the  whole  orange  do   you  give  away  ? 

Ans.  The  orange  is  divided  into  3  parts  ;  if  each  of 
these  parts  is  divided  into  4  parts,  the  whole  orange  would 
be  divided  into  12  parts,  and  each  part  is  j'^  of  the 
whole. 

1  of  1  is  J- 

5.  If  you  have  an  apple  and  it  is  cut  into  5  equal  parts, 
v;hat  part  of  the  apple  is  each  piece  ?  If  each  piece  is 
cut  into  3  equal  parts,  what  part  of  the  whole  apple  is 
each  piece  ? 

Ans.  If  an  apple  is  cut  into  5  equal  parts,  each  part  is 
onejifth  of  the  wiiole,  and  if  each  of  these  pieces  is  divi- 
ded  into  3  parts,  each  part  is  ^K  of  the  whole. 


120  ARITHMETIC.       SECOND  PART. 

6.  If  you  have  an  orange  and  it  is  divided  into  3  equal 
parts,  each  part  is  one  third,  if  each  i  is  divided  into  6 
equal  pieces,  what  part  of  the  i  is  each  piece  ? 

7.  What  part  of  the  whole  orange  is  each  piece  ? 

8.  If  a  loaf  of  hread  is  cut  into  4  equal  parts,  each  part 
is  A.  If  each  \  is  divided  into  5  equal  pieces,  each  piece 
is  i  of  the  i,  and  J^  of  the  whole  loaf,  |  of  i  then  is  J^. 

9.  If  a  sheet  of  paper  is  cut  into  5  pieces,  each  piece 
is  \.  If  each  i  is  cut  into  3  equal  pieces,  each  piece  is 
1  of  the  i,  and  y'^  of  the  whole,     i  of  i  then  is  J^. 

10.  If  a  yard  of  cloth  is  cut  into  8  equal  pieces,  and 
each  piece  is  then  cut  into  3  equal  parts,  what  part  of  the 
whole  is  each  piece  ? 

11.  If  a  bushel  of  apples  is  divided  into  fourths  of  a 
bushel,  and  each  fourth  is  divided  into  6  equal  portions, 
what  part  of  the  whole  is'each  portion  1 

12.  If  you  divide  a  pine  apple  into  3  equal  parts,  and 
each  of  those  parts  into  6  equal  pieces,  what  part  of  the 
whole  is  each  piece  ? 

13.  If  you  have  ^  of  a  dollar  and  wish  to  give  |  of  it 
to  each  of  7  children,  what  part  of  the  whole  dollar  do 
you  give  to  each  ? 

14.  If  you  have  |  of  a  lb.  of  raisins  and  wish  to  divide 
it  equally  between  3  children,  what  part  of  a  lb.  do  you 
give  to  each  ? 

15.  If  you  have  |  of  a  yard  of  muslin,  and  divide  it  in- 
to  8  equal  pieces,  what  part  of  J^  is  each  piece,  and  what 
part  of  the  irhole  yard  is  each  piece  ? 

16.  What  part  of  a  unit  is  i  of  }  ? 

Ans.  If  a  unit  is  divided  into  6  parts,  and  each  of  these 
parts  into  8,  the  unit  would  be  divided  into  48  parts,  and 
each  part  is  -^\  of  the  whole. 

Let  the  following  sums  be  stated  in  the  same  manner. 

17.  What  part  of  a  unit  is  i  of  }  ? 

18.  What  part  of  a  unit  is  i  of  |  ? 

19.  What  part  of  a  unit  is  ^  of  i  ? 

20.  What  part  of  a  unit  is  |  of  i  ? 

21.  What  part  of  a  unit  is  i  of  |  ? 

22.  What  part  of  a  unit  is  ^  of  i  ? 

23.  What  part  of  a  unit  is  \  of  ^V  - 

24.  What  part  of  a  unit  is  ^  of  ^  ? 


MULTIPLICATION  OF  VULGAR  FRACTIONS. 


121 


25.  What  part  of  a  unit  is  i  of  j\  ? 

26.  What  part  of  a  unit  is  i  of  i  1 

27.  What  part  of  a  unit  is  ^  of  y^^  ? 

28.  What  is  1  of  ^? 

29.  What  is  i  of-^V? 

30.  What  is  i  of  j\  ? 
of  1? 

siofi? 

si    Ofir? 

1  of  i 

2  0 

4  Of}? 

s  i  of  i  ? 


31.  What 

32.  What 

33.  What 
What 
What 
What 
What 
What 
What 
What 
What 
What 
What 


34. 
35. 
36. 
37. 
38. 
39. 
40. 
41. 
42. 
43. 


What  is  1  of  I  ? 
What  is  i  ofi? 
What  is  ^  of  4  ? 
What  is  1  of  i  ? 
iof^V?  What  is -pV  of  i? 
igxofi?  What  is  1  of  j? 
is  I  of  yV  ?  What  is  J^  of  i  ? 
isi  of  i?  What  is  1  oT^? 

1  of  "^V^  What  is  J- of  yV? 
s  Jj-  of  i  ?  What  is  }  of  J^  ? 
si  of  J^  ?  What  is  J-  ofi?" 
After  finding  I  of  one  third  we  know  that  i  of  two  thirds 
is  twice  as  much. 

1.  What  1  of  i  ?  What  is  i  of  |  ? 

2.  What  is  i  of  i  ?  What  is  i  of  f  ? 

3.  What  is  }  of  i  7  What  is  j  of  f  ? 

4.  What  is  i  of  i  ?  What  is  |  of  |  ? 

5.  What  is  i  of  1  ?  What  is  ^  of  f  ? 

6.  What  is  i  of  i  ?  What  is  i  of  f  ? 

7.  What  is  i  off  ?  What  is  i  off  ? 

8.  What  isi  off? 

9.  What  is  i  of  i  ?  What  is  |  of  |  ?  What  is  4  of  |  ? 
What  is  I  of  f?  What  is  j  of  f  ? 

10.  What  is  i  of  1  ?     What  is  |  of  |  ? 

11.  What  is  i  of  1  ?  What  is  I  of  f  ? 

12.  What  is  i  of  ^V 1  What  is  i  o(\\  ? 

13.  What  is  i  of  j\  ?  What  is  i  of//? 

14.  What  is  \  of  a  ?  What  is  i  of  ^  ? 

15.  What  is  i  of  I  ?  What  is  j\  of  |  ? 

16.  What  is  j\  off  ?  What  is  i  of  f  ? 

17.  What  is  i  of  /^  ?  What  is  ■^\  of /^  ? 

18.  What  is  -'  of  iJ-  ?  What  is  |  of  |f  ? 

19.  What  is  i  of  tV  ?  What  is  f  of  f  ? 

11 


122  ARITHMETIC       SECOND  PART. 

20.  What  is  1  of  I  ?  What  is  i  of  -^V  ? 

21.  What  is  i  of  i  ?  What  is  i  of  /? 

After  finding  one  part  of  a  fraction,  we  find  th&  other 
parts  by  multiplication. 

Thus  after  finding  what  one  fourth  of  a  fraction  is,  we 
can  find  three  fourths  by  multiplying  by  3. 

Thus  i  of  f  is  ^\,  therefore  f  of  f  is  3  times  as  much 
or  — ^- . 

1  .*  What  is  i  of  I  ?  What  is  |  of  f  ?  What  is  f  of  f  ? 

2.  What  is  1  of  f  ?  What  is  |  of  |  ?  What  is  |  off  ? 

3.  What  is  1  of  I  ?  What  is  f  of  |  ?  What  is  f  of  f  ? 
Let  the  pupil  reason  thus  :  What  is  |  of  |  ?     One  sixth 

of  one  third  is  Jg-.     0?ic  sixth  of  two  thirds  is  -^j.     Four 
sixth  of  two  thirds  is  4  times  as  much,  or  j\. 

4.  What  is  f  off?  Whatisioff?    What  is  |  off? 

5.  What  is  f  of  f  ?  What  is  f  of  f  ?  What  is  f  of  4  ? 
What  is  f  of  I  ?  What  is  ^  of  #  ? 

6.  What  is  |  of  ^%  1  What  is  -^  of  j-i  ? 

7.  What  is  H  of|?  What  is  ]4  off? 

8.  What  j%  off  ?  What  is  |  of  f  ?  What  |  of  f  ?  What 
is  f  of  I  ?  What  is  f  of  I  ? 

in  multiplying  one  fraction  by  another,  we  are  to  take 
a  certain  part  of  one  fraction,  as  often  as  there  are  units 
in  the  numerator  of  the  other  fraction. 

Thus,  if  we  are  to  multiply  |  by  ^  we  are  to  take  a  sixth 
of  I  four  times. 

To  explain  the  rule  for  multiplying,  when  both  factors 
are  fractions,  take  an  example. 

What  is  I  of  I  ? 

One  fifth  off  is  3*^^,  and  this  is  made  by  multiplying  the 
denominator  6,  by  the  denominator  5. 

Three  fifths  of  |  is  three  times  as  much  or  if,  and  this 
is  made  by  multiplying  the  numerator  4,  by  the  numera- 
tor 3. 

Therefore  multiplying  the  denominators  together  ob- 
tained one  fifth  off,  and  multiplying  the  numerators  iogQih- 
er,  obtained  three  fifths. 


simple  division.  123 

Rule   for  Multiplying  when   both   factors   are 
fractions. 

Multiply  tJie  denominators  together  to  ohtain  one  fart,  and 
the  numerators  together  to  obtain  the  required  number  of 
parts. 

In  performing  these  sums  upon  the  slate,  let  the  pupil 
use  the  signs  thus  : 

Multiply  I  by  ^\.     f  X  fV  =  hi 

Exercises  for  the  Slate. 

What  is  J^  of  fV  ?  What  is  f  of  f  ?     What  is  f  of  |  ? 
What  is  f  of  I  ?"What  is  ^  of  Jf  ?  What  is  -,%  of  ff  ^ 
What  is  If  of  f  I  ?     What  is  =^ «  of  Hi  ?    What  is  |H 

of  S-i-fi-S  ? 

Multiply  H  by  ^  ^     Multiply  ||f  by  ^Vo • 
Multiply  \l\  by  m-     Multiply  if^  by  \^. 


DIVISION. 

Division  is  finding  how  often  one  number  is  contained 
in  another,  and  thus  finding  what  part  of  one  number  is 
another  number. 

The  number  to  be  divided  is  called  the  Dividend. 

The  number  by  which  we  divide  is  called  the  Divisor. 

The  answer  is  called  the  Quotient. 

What  is  left  over,  after  division,  is  called  the  Remainder. 

There  are  four  kinds  of  division. 

The  first  is  Simple  Division,  in  which  both  the  dividend 
and  divisor  are  whole  numbers,  and  ten  units  of  one 
order,  make  one  unit  of  the  next  higher  order. 

The  second  is  Compound  Division,  in  which  other  num. 
bers  besides  ten,  make  units  of  higher  orders. 

The  third  is  Division  of  Vulgar  Fractions,  in  which  the 
dividend  or  divisor  (or  both)  are  Vulgar  Fractions. 

The  fourth  is  Decimal  Division,  in  which  the  dividend, 
or  divisor,  (or  both)  are  decimal  fractions. 


124  ARITHMETIC.       SECOND  PART, 

SIMPLE  DIVISION. 

How  many  9  cents  are  there  in  63  cents  ? 

What  part  of  63  cents  is  9  cents  1 

How  many  times  is  8  contained  in  56  ? 

If  8  is  contained  7  times  in  56,  what  part  of  56  is  8  ? 

If  56  is  divided  by  8,  how  much  smaller  is  the  quotient 
than  the  dividend  ? 

How  many  7  dollars  are  there  in  42  dollars  ? 

What  part  of  42  dollars  is  7  dollars  ? 

How  many  times  is  6  contained  in  66  ? 

If  6  is  contained  11  times  in  66,  what  part  of  66  is  6? 

If  66  is  divided  by  6,  how  much  smaller  is  the  quotient 
than  the  dividend  ? 

There  are  many  numbers  which  cannot  be  divided  into 
equal  parts,  without  making  a  fraction.  For  example,  if 
we  wish  to  divide  7  apples  into  two  equal  portions,  we 
should  have  for  answer  3  apples  and  i  of  an  apple. 

If  we  had  13  apples,  and  wished  to  give  a  third  of  them 
to  each  of  3  friends,  we  should  divide  the  13  by  3,  and  the 
answer  would  be  4,  and  1  left  over.  That  is,  we  could 
give  4  apples  to  each  of  the  3  friends,  and  one  would  be 
left  to  divide  among  them.  This  divided  by  3,  (or  into  3 
equal  parts)  would  give  a  third  to  each  one.  13  then,  di- 
vided  by  3,  gives  4  and  ^  as  answer. 

If  you  are  to  divide  7  apples  equally  among  3  persons, 
how  many  whole  apples  would  you  give  to  each,  and  what 
would  remain  to  be  divided  ? 

If  you  had  14  oranges,  and  wished  to  divide  them  equal- 
ly among  6  persons,  how  many  whole  oranges  would  you 
give  each  ? 

How  v\;ould  you  divide  the  two  that  remained  ? 

Ans.  Divide  each  into  6  equal  parts,  and  give  one  of 
the  parts  of  each  orange  to  the  6  persons.  Each  person 
would  then  have  2  oranges  and  |. 

If  you  have  two  apples,  each  cut  into  12  parts,  and  take 
4  of  these  parts  from  each  apple,  how  much  do  you  take  ? 

Ans.  j%.  For  j\  from  each  apple  makes  /^  in  the  whole. 

If  we  take  9  twelfths  from  eacii  of  the  two  divided  ap- 
pies,  we  shall  have  ||  in   the  whole. 

Now  this  is  not  if  of  one  apple,  for  nothing  has  more 


SIMPLE    DIVISION. 


125 


than  12  twelfths.   Whenever  therefore  we  find  an  improper 
fraction,  we  know  that  more  than  one  unit  has  been  divided. 

What  part  of  13  apples  is  3  apples  and  \  of  an  apple  ? 

Ans.  It  is  a  fourth  of  13,  because  4  times  3  and  ^  make 
13. 

What  part  of  5  is  1  ?  is  2  ?  is  3  ?  is  4  ?  is  6  ? 

In  the  last  question  we  reason  thus  :  If  1  is  one  fifth  of 
5,  6  must  be  6  times  as  much,  or  f  of  5. 

What  part  of  8  is  1  ?  is  4  ?  is  7  ?  is  9  ? 

What  part  of  15  is  1  ?  is  2  ?  is  3  ?  is  14  ?  is  19  ? 

What  part  of  10  is  1  ?  is  2  ?  is  5?  is  9?  is  11  ?  is  20  ? 

What  is  a  sixth  of  19  1  What  is  a  fourth  of  21  ? 

What  is  an  eighth  of  26? 

If  you  had  19  pears,  and  divided  them  equally  among 
6  persons,  how  much  did  you  give  to  each  ? 

What  part  of  19  is  3  and  i. 

Ans.  As  there  is  6  times  3  and  ^  in  19,  it  is  i  of  19. 

When  one  number  is  placed  over  another,  it  signifies 
that  the  upper  number  is  divided  by  the  lower. 

Thus,  I  signifies  that  the  3  is  divided  by  4.  For  a 
fourth  of  three  things  is  3  fourths,  and  f  signifies  either 
3  fourths  of  one  thing,  or  a  fourth  of  3  things. 

If  you  wish  to  divide  3  dollars  into  5  equal  parts,  what 
would  it  be  necessary  to  do,  before  you  could  divide  them  l 
Ans.  Change  them  to  dimes. 

What  would  be  the  answer  ? 

If  you  wished  to  divide  4  dimes  into  10  equal  parts, 
what  would  it  be  necessary  to  do  before  you  could  divide 
them? 

What  would  be  the  answer? 

(Let  this  be  shown  by  the  coins.) 

How  can  3  dollars  be  divided  so  as  to  give  ten  of  the 
class,  each  an  equal  part  ? 

Ans.  Change  the  dollars  to  dimes,  and  then  dividing 
them  into  ten  equal  parts,  there  will  be  3  dimes  for  each 
of  the  ten. 

Divide  ^1,2  so  as  to  give  6  scholars,  each  an  equal  part. 

Divide  $2,4  so  as  to  give  8  scholars,  each  an  equal 
part  ? 

Divide  1  dime  equally  between  two  scholars. 

Divide  1  dime  5  cents  equally  between  3  scholars. 
11* 


126  ARITHMETIC.       SECOND    PART. 

If  1  dime  8  cents  are  divided  by  6,  what  is  the  answer  ? 
If  3  dimes  9  cents  are  divided  by  6,  what  is  the  quo- 
tient, and  what  the  remainder  ? 

If  5  dimes  6  cents  are  divided  by  7,  what  ate  the  quo- 
tient and  remainder? 

If  4  dimes  7  ceots  are  divided  by  6,  what  are  the  quo- 
tient and  remainder  ? 

In  the  above  sums,  it  will  be  seen  that  when  one  order  of 
the  dividend  will  not  contain  the  divisor  once,  it  is  reduced, 
and  added  to  the  next  lower  order,  and  then  divided. 

Thus  when  4  dimes,  6  cents  were  to  be  divided  by  6,  the  4 
dimes  were  changed  to  cents,  and  added  to  the  6  cents, 
and  then  divided. 

It  will  also  be  seen,  that  the  quotient  and  the  remainder 
are  always  of  the  same  order  as  the  dividend. 

Thus  if  4  dimes  7  cents  are  divided  by  6,  the  4  dimes 
are  reduced,  and  added  to  the  cents,  and  the  quotient  is  7 
cents,  and  the  remainder  is  5  cents. 

Thus,  also,  if  17  thousands  are  divided  by  5,  the  quo- 
tient is  3,  and  2  remainder.  The  3  is  3  thousands,  and  the 
2,  is  2  thousands. 

If  the  order  of  the  dividend  were  millions,  the  quotient 
and  remainder  would  also  be  millions. 

If  the  order  were  tens  the  quotient  and  remainder 
would  also  be  tens. 

If  we  divide  8  tens  by  3,  the  quotient  is  2  tens,  and  the 
remainder  2  tens. 

When  the  dividend  has  several  orders,  we  divide  each 
order  separately,  beginning  with  the  highest  orders.  This 
is  called  Short  Division. 

If  there  is  any  remainder,  after  the  division  of  each  or- 
der, it  is  changed  to  the  next  lower  order,  added  to  it,  and 
then  divided. 

For  example.     Let  9358  be  divided  by  4. 
We  first  divide  the  9  thousands  by  4,  add  the   remain- 
der to  the  3  hundreds  and  divide  that.     Then   divide  the 
tens  and  units. 

Place  them  thus  :         4)9358 

2339a 


SIMPLE   DIVISION.  127 

The  9  thousands  is  first  divided.  In  9  iinits  there  would 
bo  2  fours,  and  1  remainder.  But  as  this  is  9  thousands, 
the  quotient  and  remainder  must  be  the  same  order  as  the 
dividend,  and  the  2,  is  2  thousand  fours,  and  is  set  under 
the  9  in  the  thousands  order.  The  remainder  also  is  1 
thousand,  and  is  changed  to  hundreds  and  added  to  the  3, 
making  it  13  hundred.  This  is  then  divided  by  4.  The 
quotient  is  3  hundreds,  which  is  put  under  that  order,  and 
the  1  hundred  that  remains,  is  changed  to  tens  and  added 
to  the  5  tens,  making  15  tens.  This  is  divided  by  4,  and 
the  quotient  is  3  tens,  which  is  set  in  that  order.  3  tens 
remain  which,  changed  to  units  and  added  to  the  8,  make 
38  units.  This  is  divided  by  4,  and  the  quotient  is  9  units, 
which  is  put  in  that  order.  2  units  remain,  which  are  di- 
vided by  the  4  thus  f . 

93.58,  then,  contains  4,  2  thousands  of  times,  3  hundreds 
of  times,  3  tens  of  times,  and  9  units  of  times.  The  3  left 
over,  is  |  of  another  time. 

Let  the  pupil  in  performing  each  operation  on  the  slate, 
explain  it  thus  : 

7)2496 


356  4 

7  is  contained  in  24  units  3  times,  in  24  hundreds,  3 
hundred  times,  which  are  set  in  the  order  of  hundreds. 
3  hundred  are  left  over,  wiiich  changed  and  added  to  the 
9  tens,  make  39  tens. 

7  is  contained  in  39  tens,  5  tens  of  times,  which  are  set 
in  the  order  of  tens.  4  tens  are  left  over,  which,  changed 
and  added  to  6,  make  46  units. 

Divide  46  units  by  7,  and  the  answer  is  6  units,  which 
are  set  in  that  order,  and  4  remain,  which  have  the  7  set 
under  them,  to  show  that  they  are  divided  by  7. 


Rule  for  Short  Division. 

Divide  the  highest  order,  and  set  the   quotient  under  it. 

If  any  remains,  reduce  and  add  it  to  the  next  lower  order, 

and  divide  as  before.     If  the  number  in  any  order,  is  less 

than  the  divisor,  place  a  cipher  under  it  in  the  quotient ;  then 


128 


ARITHMETIC.       SECOND  PART. 


reduce  and  add  it  to  the  next  lower  order,  and  divide  as  be- 
fore. If  any  remains  when  the  lowest  order  is  divided, 
place  the  divisor  under  it  as  a  fraction. 


Divide 


Examples. 


3694  by 

3 

Divide       3456  by  3 

4329  " 

4 

7892  "  4 

6548  « 

5 

3456  «  5 

3621  « 

6 

7892  "  6 

4638  « 

7 

1234  «  7 

29639  " 

8 

5678  «   8 

36964  « 

9 

91234  «   9 

24697  " 

10 

56789  «  10 

36941  « 

11 

12345  «  11 

1263  " 

12 

67891  "  12 

When  loth  the  divisor  and  dividend,  have  several  orders, 
another  method  is  taken  called  Long  Division.  Let 
6492  be  divided  by  15.  In  performing  the  operation  de- 
scribed below,  we  set  the  figures  thus. 

Dividend. 

Divisor  15)6492(432  if  Quotient. 
60 

49 
45 


42 
30 

I2 

We  first  take  as  many  of  the  highest  orders  as  would, 
if  units,  contain  the  divisor  once  and  not  more  than  9 
times.  In  this  case  we  take  64  hundreds.  Now  we  can- 
not very  easily  find  exactly  how  many  times  the  15  is  con- 
tained in  64  hundreds.  But  we  can  find  how  many  hun- 
dreds of  times  it  is  contained  thus.  As  15  would  be  con- 
tained 4  units  of  times,  in  64  units,  it  is  contained  4  hun- 
dreds of  times,  in  64  hundreds.  Which  400  is  to  be  set 
in  the  quotient,  (omitting  the  ciphers.) 

As  we  have  found  that  the  dividend  contains  15,  4  hun- 
dreds of  times,  we  subtract  4  hundred  times  15  from  the 
dividend,  to  find  how  often  15  is  contained  in  what  re- 


SIMPLE    DIVISION.  129 

mains.  400  times  15  is  60  hundreds  (6000)  which  sub- 
tracted from  the  64  hundreds,  leaves  4  hundreds. 

This  4  hundreds  changed  to  tens,  and  the  9  tens  of  the 
dividend  put  with  it,  make  49  tens.  We  now  find  how 
many  tens  of  times  the  15  is  contained  in  the  49  tens,  thus  : 
as  15  would  be  contained  3  units  of  times  in  40  units,  it  is 
contained  3  tens  of  times  in  49  tens,  which  3  tens  is  set  in 
the  quotient.  Wcnow  subtract  3  tens  of  15  (or  45  tens) 
from  the  49  tens,  and  4  te7is  remain.  These  are  changed 
to  U7iits  and  have  the  2  units  of  the  dividend  pnt  with 
them,  making  42  units.  15  is  contained  in  42  units  2 
units  of  times,  which  is  set  in  the  quotient.  Twice  15 
from  42  units,  leave  12,  which  is  J-^  of  another  15,  The 
15  then,  is  contained  in  the  dividend,  4  hundreds  of  times, 
3  tens  of  times,  2  units  of  times,  and  ||  of  another  time, 
or  432  times,  and  ||  of  another  time. 

Again,  divide  0998  by  24. 

To  do  it  we  first  find  how  many  hundreds  of  times,  the 
dividend  contains  the  divisor,  and  subtract  these  hundreds  ; 
then  how  many  tens  of  times,  and  subtract  these  tens  ; 
then  how  many  miits  of  times  and  subtract  these  units  ; 
and  then  what  remains  has  the  divisor  set  under  it. 

Let  the  pupil  in  doing  sums  explain  them  as  below. 

24)fi998(291iA 

48 

219 
216 

38 
24 

14 

24  is  contained  in  69  zmits,  2  times ;  in  69  hundreds,  2 
hundred  times.  2  hundred  times  24  is  48  hundred,  which 
subtracted  from  69  leaves  21  hundred. 


130  '  ARITHMETIC.       SECOND  PART, 

21  hundreds  are  210  tens,  and  the  9  tens  of  the  dividend 
brought  down,  make  219  tens. 

24  is  contained  in  219,  9  times  ;  in  219  tens,  9  tens  of 
times.  24  multiplied  by  9  tens,  is  216  tens,  which  sub- 
tracted  from  219  tens  leaves  3  tens. 

3  tens  are  30  units,  and  the  8  units  of  the  dividend 
brought  down  make  38  units.  24  is  contained  in  38 
units  once,  and  14  over,  which  is  4^|  of  another  time. 

The  dividend  then  contains  the  divisor  2  hundreds  of 
times,  9  tens  of  times,  1  unit  of  times,  and  if  of  another 
time,  or  291  times  and  if  of  another  time. 

Thus  it  appears,  that  in  Long  Division,  each  quotient 
figure,  when  set  down,  does  not  show  the  exact  number  of 
times  the  divisor  is  contained  in  the  order  which  is  divided ; 
but  it  shows,  that  the  divisor  is  contained  so  many  times 
as  the  quotient  figure  expresses,  and  then,  a  process  fol- 
lows for  discovering  how  many  more  times  it  is  contained. 

Let  the  pupil  do  the  following  sums,  and  explain  them 
as  above,  until  perfectly  familiar  with  the  mode. 

Divide     2479  by  14  Divide     3568  by  16 

1954  "  18  "          5896  "  23 

"       36964  "  17  "        38907  "  21 

"       29006  "  28  "       46032  "  36 

Rule  foe  Long  Division. 

Place  the  divisor  at  the  left  of  the  dividend,  and  draw  a 
line  between.  Take  as  many  of  the  highest  orders  as  would, 
if  units,  contain  the  divisor  once,  and  not  more  than  9 
times.  Divide  the  orders  so  taken,  as  if  they  were  units. 
Place  the  quotientfigure  at  the  right  of  the  dividend,  and  draw 
a  line  beticeen.  Multiply  the  quotient  and  the  divisor  to. 
gether,  and  subtract  them  from  the  -part  of  the  dividend  al- 
ready divided.  To  the  remainder,  add  as  many  of  the  next 
undivided  orders  of  the  dividend  as  ivould  enable  it,  if  units, 
to  contain  the  divisor  once,  and  not  more  than  9  times,  and 
then  divide  as  before. 

If  it  is  needful  to  add  more  than  one  order  of  the  dividend 
to  any  remainder,  {to  enable  it  to  contain  the  divisor)  put  one 
cipher  in  the  quotient  for  every  additional  order.     If  any 


SiaiPLE  DIVISION.  131 

remains  after  dividing  the  unit  order,  put  the  divisor  under 
it  for  a  fraction. 


Examples. 

Divide  2649 

by 

12 

Divide 

3294 

by 

14 

2468 

(( 

16 

ti 

64329 

le 

16 

1234 

(< 

17 

t< 

5678 

li 

18 

56789 

ti 

19 

K 

8234 

le 

36 

35073 

cc 

59 

tl 

76542 

tc 

41 

45078 

a 

256 

(C 

96743 

(( 

348 

912345 

(( 

481 

(C 

59624 

« 

562 

678122 

(( 

984 

(( 

23864 

(( 

541 

34508 

<c 

639 

fC 

35469 

(C 

856 

543219 

<( 

656 

(( 

1459862 

(( 

942 

678912 

(( 

9481 

1( 

724368 

a 

2586 

9876533 

a 

6002 

(< 

159864 

i( 

2851 

Examples  foe  Mental  Exercises. 

1.  Bought  12  pounds  of  raisins  for  3  shillings  a  pound, 
how  many  dollars  did  they  cost  ? 

State  the  process  thus.  If  one  pound  cost  3  shillings, 
12  pounds  cost  12  times  as  much,  or  36  shillings.  As 
there  are  6  shillings  in  a  dollar,  they  cost  as  many  dollars 
as  there  are  sixes  in  36. 

Let  the  following  sums  be  stated  in  the  same  manner. 

2.  Bought  5  bushels  of  peaches  at  4  shillings  a  bushel, 
how  many  dollars  did  they  cost  ? 

3.  How  many  peaches  at  4  cents  each  must  you  give 
for  9  oranges  at  5  cents  apiece. 

State  the  last  sum  thus.  If  one  orange  cost  5  cents,  9, 
cost  9  times  as  much,  or  45  cents.  As  each  peach  is 
worth  4  cents,  you  must  give  as  many  peaches  as  there 
^iQ  fours  in  45. 

4.  If  you  buy  10  yards  of  cotton,  at  5  shillings  a  yard, 
and  pay  for  it  with  butter  at  2  shillings  a  pound,  how  ma- 
ny pounds  will  pay  for  it  ? 

5.  How  many  apples  at  4  cents  each,  must  you  give  for 
3  pine  apples  at  12  cents  each  ? 

6.  If  you  buy  48  bushels  of  coal  for  12  cents  per  bush- 


132  ARITHMETIC.       SECOND   PART. 

el,  and  pay  for  it  with  cheese  at  10  cents  per  lb.  how  ma- 
ny pounds  do  you  give  ? 

7.  How  much  rye  at  5  shillings  a  bushel  must  you  give 
for  12  bushels  of  wheat  at  8  shillings  a  bushel. 

8.  How  much  cloth  worth  9  shillings  a  yard  must  you 
give  for  a  firkin  of  butter  worth  12  dollars  ? 

(Change  the  dollars  to  shillings.) 

9.  How  many  dozen  of  eggs  at  9  cents  per  dozen  must 
be  given,  for  3  yards  of  cotton  worth  20  cents  per  yard? 

10.  If  you  have  8  pine  apples  worth  9  cents  each,  and 
your  companion  has  9  quarts  of  strawberries  worth  8  cents 
a  quart,  which  he  gives  to  buy  the  same  worth  of  pine  ap- 
ples, how  many  pine  apples  must  you  give  him  ? 


COMPOUND  DIVISION. 

Divide  £4  „  85.  by  2. 

Divide  £6  „  12s.  by  3. 

If  2  dresses  contain  24  }  ds.  2  qrs.  how  much  in  each 
dress  ? 

If  3  silver  cups  weigh  9  lbs.  6  oz.  what  is  the  weight 
of  each  ? 

In  division  we  find  how  often  one  number  is  contained 
in  another,  and  thus  what  part  of  one  number  is  another. 
Thus  if  we  divide  8  lbs.  16  oz,  by  4,  we  can  either  say 
how  many  times  is  4  contained  in  8  and  in  16,  or  we  can 
say  what  is  one  fourth  of  8  lbs.  and  16  oz. 

If  there  is  any  remainder  in  dividing  one  order,  it  must 
be  changed  to  units  of  the  next  lower  order  and  added  to 
it  and  then  divide  again. 

In  doing  the  sum  we  place  the  figures  thus. 

£  5.  d. 

3)4   «    18     "     9 


1    ''    12     "    11 

We  proceed  thus  in  explaining  the  process. 

A  third  of  4£  is  1£  which  is  set  under  that  order,  and 
there  is  1£  remaining  which  is  changed  to  20  shillings 
and  added  to  the  18  making  38.     A  third  of  38  shillings  is 


SIMPLE   DIVISION.  133 

12  shillings,  which  are  set  under  that  order.  2  shillings 
remain  which  are  changed  to  24  pence  and  added  to  the 
9d,  making  33  pence  ;  a  third  of  33  pence  is  11  pence 
which  are  set  in  that  order. 

Let  the  following  sums  be  performed  and  explained  as 
above. 

Divide  22i;  11*.  6d.  by  6. 

At  2£  8s.  6(1.  for  6  pair  of  shoes  what  is  that  a  pair  ? 

If  9  silver  cups  weigh  31b.  6oz.  8pwt,  3qr.  what  is  the 
weight  of  each  ? 

If  8  dresses  contain  59  yds.  3  qrs.  2n.  how  much  in 
each  dress  ? 

If  the  divisor  exceeds  12  and  is  a  composite  number  di- 
vide the  sum  by  one  of  the  factors  as  above  and  the  an- 
swer by  the  other. 

Examples. 

Divide  2£  «  8s.  «  lid.  "  4qr.  by  44. 

If  18  gal.  "  Cqr.  "  4g.  of  brandy  be  divided  equally 
into  28  bottles  how  much  does  each  contain  ? 

If  24  coats  contain  62  yds.  3  qrs.  4  na.  how  much  does 
each  contain  ? 

If  32  teams  be  loaded  with  40T.  16  cwt.  3  qrs.  how 
much  is  that  for  each  team  ? 

If  the  divisor  exceeds  12  and  is  not  a  composite  number 
the  following  method  is  used. 

Let  the  figures  be  placed  thus. 

j£.         s.         d.     £.     s.       d. 
139)461  "  11  "  11(   3"    6  "    5 
417 

44 

20 

891 
834 

*  «7 

12 


695 
695 


12 


134  ARITHMETIC.       SECOND  PART. 

We  first  divide  the  pound  order  and  3  is  the  quotient 
tigure,  which  is  of  the  pound  order  because  the  dividend  is 
pounds.  This  is  put  in  the  quotient  with  the  £  put  over 
it  to  indicate  its  order. 

In  order  to  find  the  remainder  we  subtract  the  product 
of  the  quotient  and  divisor  from  the  461. 

The  remainder  is  44£.  This  must  be  changed  to 
sliillings  which  is  done  by  multiplying  it  by  20  and  then 
the  11  shilHngs  of  the  dividend  are  added. 

This  sum  is  then  divided  by  139  and  the  quotient  fig- 
ure is  6,  which  is  of  the  shilling  order  and  must  be  put  in 
the  quotient  under  that  sign.  Proceed  as  before  till  the 
orders  are  all  thus  divided. 

Let  the  following  examples  be  performed  and  explain- 
ed as  above. 

Divide  239£  "  16s.   "  4d.   "  Sqr.  by  123. 

If  239  yds.  of  cloth  cost  49^  195.  lid.  what  was  that 
per  yard  ? 

Note. — Change  the  pounds  to  shillings  first. 

If  349  cwt.  3  qrs.  12  lbs.  is  contained  in  264  barrels 
how  much  is  in  each  barrel. 

If  42  cwt.  of  tobacco  cost  826^  18s.  9d.  what  is  that 
per  lb. 


Rule  for  Compound  Division. 

If  the  divisor  does  not  exceed  12,  divide  each  order  sepa- 
rately, beginning  with  the  highest,  rememhering  to  make  the 
quotient  figure  of  the  same  order  as  the  dividend. 

Whenever  there  is  a  remainder  change  it  and  add  it  to 
the  next  lower  order  and  divide  as  before. 

If  the  divisor  exceeds  12,  either  resolve  it  into  factors  and 
divide  first  by  one  and  then  by  the  other,  or  proceed  after 
the  manner  of  long  Division. 


T.         cwt. 

lb. 

oz.         dr. 

Divide 

29     "    13 

"    25 

"    12    "    13  by  6. 

lb.           oz. 

pwt. 

grs. 

Divide 

7      "     10 

"    15 

"     2  by  5. 

niVISION  OF  VULG.VR  FKACTIONS.  135 

yds.  qrs.  na. 

Divide      76  "3  "2  by  4. 

deg.  m.  fur.       pol.        Jt.       in.  bar. 

Divide      97  "    55  "     7      »    35    «'    4     "  2       1 
by  7. 

£  s.  d.         qrs. 

Divide     .25  "    16  "     10    "     3    by  9. 


Division  where  only  the  Divisor  is  a  Fraction. 

If  we  have  3  apples,  how  many  ^  in  the  whole  ?  Ans. 
In  one  apple  there  are  two  halves,  and  in  three  apples  there 
are  three  times  as  many,  or  s/x  halves. 

If  we  have  6  dollars  how  many  i  in  the  whole  ?  Ans, 
In  one  dollar  there  are  three  thirds  and  in  six  dollars 
there  are  six  times  as  many,  or  eighteen  thirds. 

If  we  have  9  apples  how  many  i  ? 

In  8  apples  how  many  J-  ? 

In  12  apples  how  many  |  ? 

In  7  apples  how  many  j\  1 

It  thus  appears  that  when  we  divide  by  a  fraction  (un- 
less it  be  an  improper  fraction)  the  quotient  is  larger  than 
the  dividend. 

Thus  12  divided  by  \  is  48,  for  there  are  48  onejourths 
in  12  units. 

Again  9  divided  by  i  is  27,  for  there  are  27  one  thirds 
in  9  units. 

How  many  ^  in  8  ? 

How  many  ^  in  12  ? 

Divide  7  by  \  Divide  6  by  i  Divide  12  bv  i  Divide 
10  by  i  Divide  8  by  i  Divide  11  by  i  Divide  12  bv 
j\    Divide  9  by  i.        " 

How  many  i  in  8  ?  How  many  i  in  7  ? 

If  you  divide  8  by  1  the  answer  is  24,  for  there  are  24 
one  thirds  in  8.  But  if  we  are  to  divide  8  by  |  there  will  be 
but  half  as  many.  For  there  is  but  half  as  many  two 
thirds  as  there  one  thirds  in  a  number.  Therefore  if  H  di- 
vided by  i  is  24,  when  divided  by  |  it  is  half  as  much, 
or  12. 

How  many  |  in  3  ? 


136  ARITHMETIC.       SECOND    PART. 

Ans.  \n  3  there  are  18  o7ie  sixths  and  half  as  many  tuv 
sixths,  or  9. 

How  many  f  in  12  ? 

How  many  f  in  2  ? 

How  many  ^  in  4  ? 

How  many  f  in  6  ? 

How  many  |  in  3  ? 

Divide  4  by  f  Divide  5  by  §  Divide  3  by  f  Divide 
8  by  I  Divide  2  by  f  Divide  7  by  f  Divide  5  by  j% 
Divide  12  by  |. 

If  you  have  12  yards  of  long  lawn  and  wish  to  cut  a 
number  of  handkerchiefs  of  f  of  a  yard  each,  how  many 
can  you  make  from  the  whole  piece  ? 

If  you  have  4  oranges  and  wish  to  give  |  of  an  orange 
to  your  mates,  to  how  many  could  you  give  them  ? 

If  you  have  4  pounds  of  rice  to  distribute  to  the  poor, 
and  are  to  give  f  of  a  pound  to  each  person,  to  how  many 
persons  can  you  give  1 

If  a  reservoir  is  filled  by  a  spout  in  ^  of  an  hour,  how 
many  limes  would  the  cistern  be  filled  in  9  hours? 

If  a  pound  of  raisins  can  be  bought  for  f  of  a  dollar, 
how  many  pounds  can  you  buy  for  4  dollars  ? 

If  I  of  a  barrel  of  flour  will  last  a  family  one  week, 
how  long  vtill  6  barrels  last  ? 

If  a  cow  eats  |  of  a  ton  of  hay  a  month,  how  long 
would  4  tons  last  her  ? 

if  f  of  a  barrel  of  flour  last  a  family  one  week,  how 
long  will  10  barrels  last  ? 

It  is  seen  by  the  preceding  examples,  that  when  a  num- 
ber is  to  be  divided  by  a  fraction,  it  is  multiplied  by  its  de- 
nominator, and  divided  by  its  numerator. 

Thus  if  we  are  to  divide  2  by  |  we  multiply  by  the  de- 
nominator  4  to  change  2  into  Jourths  and  then  divided  by 
the  3  to  find  how  many  three  fourths  there  are. 

Divide  3  by  |. 

Why  do  you  multiply  by  the  denominator  ?  Why  do 
you  divide  by  the  numerator  ? 


division  of  vulga.r  fractions.  137 

Rule  for  Fractional  Division  where  only  the  di- 
visor IS  A  fraction. 

Multiply  the  dividend  hy  the  denominator,  and  divide  the 
product  by  the  numerator, 

EXAMPLES  FOR  THE  SLATE. 

Divide         23     by     4  Divide 


23 

by 

25 

(( 

32 

(( 

325 

cc 

9479 

(C 

342 

(( 

681 

(( 

3292 

(( 

364 

by 

24 

21 

486 

381 

542 

232 

4285 

M 

H 

2  1  i 

n 

i^ 

Division  where  the  Dividend  only  is  a  Fraction'. 

When  the  dividend  only  is  a  fraction,  and  we  divide  it 
by  a  whole  number,  we  are  to  find  how  many  parts  oj  a 
time,  a  certain  number  is  contained  in  certain  parts  of  a 
unit. 

Thus  if  we  divide  \  by  2,  we  know  that  \  does  not  con- 
tain  2  units  once,  but  we  can  find  what  'part  of  one  time  the 
i  contains  the  2. 

If  1  is  divided  by  1  unit,  we  find  that  it  contains  it,  not 
once,  but  i  of  once.  It  can  contain  two  units  but  half  as 
many  times  as  one  unit.  Therefore  \  contains  1  one  half 
a  time,  and  it  contains  2  just  luilf  as  often,  or  ^  of  a  time.  | 
divided  by  2  then  is  \.  If  i  is  to  be  divided  by  3,  we 
reason  in  the  same  way.  \  contains  1,  i  a  time. 
It  contains  3  only  a  third  as  often,  i  of  ^  is  |,  and  there- 
fore i  contains  3,  i  of  a  time. 

Again  if  \  is  to  be  divided  by  4,  we  reason  thus  : 

If  i  contains  1,  i  a  time,  it  contains  4  only  {  as  often. 
^  of  i  is  i.  Then  i  contains  4  not  one  time,  but  \  of  one 
time. 

Again  let  i  be  divided  by  4,  and  we  reason  thus  : 

If  i  is  divided  by  1,  it  contains  it  not  1  time,  but  i 
12* 


138  ARITHMETIC.       SECOND  PART. 

of  one  time.  But  it  can  contain  4  only  i  as  often,  i  of 
i  is  -j-'^. 

The  dividend  i  contains  the  divisor  4,  not  one  time,  but 
y\  of  one  time. 

Divide  1,  i,  |,  1,  i,  ,V,  each  by  one. 

Proceed  thus  :  i  contains  1  not  one  time,  but  i  of  one 
time.     I  contains  1  not  one  time,  but  ^  of  one  time,  Aic. 

Divide 


by  2 

Divide 

i    by 

3 

u   4 

(< 

T       " 

5 

"   6 

(< 

1       " 

Q 

10 

"  7 

(( 

1       a 

]  0 

7 

"   8 

<( 

_>_     " 

9 

How  often  is  2  contained  in  i  ? 

(Ans.)  As  1  would  be  contained  i  of  a  time,  2  is  con- 
tained fialfas  often,  or  Jg  of  one  time- 
How  often  is  3  contained  in  |  ?  *' 

How  often  is  5  contained  in  i  ? 

How  often  is  6  contained  in  i  ? 

How  often  is  7  contained  in  -J^  ? 

How  often  is  8  contained  in  J,-  ? 

How  often  is  9  contained  in  j^l 

How  often  is  10  contained  in  i  ? 

How  often  is  11  contained  in  i  ? 

How  often  is  12  contained  in  ^  ? 

How  often  is  9  contained  in  i  ?  • 

How  often  is  8  contained  in  |  1 

How  often  is  9  contained  in  ^  ? 

After  finding  liow  often  4  is  contained  in  one  part,  we 
find  by  multiplying,  liow  often  it  is  contained  in  a  given 
7iumher  of  parts.  For  instance,  4  is  contained  in  one  fifth 
-J^  of  «?je  time.  In  two  fiftlis  it  would  be  contained  twice 
as  often,  or  -^^  of  one  time. 

Again,  let  s^  be  divided  by  4,  and  we  reason  thus  :  4  is 
contained  in  one  seventh  one  fourteenth  of  one  time,  in  2 
sevenths  it  is  contained  twice  as  often,  or  two  fourteenths 
of  one  time. 

How  often  is  3  contained  in  f  ? 

(Ans.)  3  is  contained  in^  one  eighteenth  of  one  time. 
In  A  it  is  contained  4  times  as  often,  or  four  eighteenths 
of  one  time. 


DIVISION    OF    VULGAR    FRACTIONS.  139 

How  often  is  4  contained  in  f  ? 

How  often  is  5  contained  in  f  ? 

How  often  is  6  contained  in  |^  ? 

Divide  -y'o  ^Y  ^-     Divide  ^  by  5. 

Divide  |  by  6.     Divide  f  by  7. 

Divide  j\  by  8.     Divide  f  by  9. 

Divide  |  by  11.     Divide  |  by  8. 

How  many  times  is  6  contained  in  ^  ? 

How  many  times  is  4  contained  in  |  ? 

How  many  times  is  7  contained  in  |  ? 

How  many  times  is  8  contained  in  |  ? 

How  many  times  is  9  contained  in  -^^  ? 

In  all  the  above  cases  it  will  be  observed  that  the  an- 
swer is  obtained  by  simply  multiplying  the  denominator  of 
the  fraction  by  the  divisor. 

Thus  f  is  divided  by  4  thus.  4  is  contained  in  ^  Jy  of 
one  time,  and  in  f  twice  as  often,  or  ^\  of  one  time.  It 
can  be  seen  that  the  answer  is  obtained  by  multiplying  the 
denominator  of  f  by  the  divisor  4.  This  is  a  method 
which  can  always  be  pursued  in  dividing  any  fraction  by  a 
whole  number,  viz  :  "  multiply  the  denominator  by  the  di- 
visor." 

But  there  is  another  method  which  is  sometimes  more 
convenient. 

Let  -^g  be  divided  by  4, 

Now  the  quotient  of  8  units  divided  by  4,  is  2  units.  Of 
course  the  quotient  of  8  sixteenths  divided  by  4,  is  2  six- 
teenths. In  this  case  we  have  divided  the  numerator  by 
the  divisor  4.  This  can  be  done  in  all  cases  where  the 
numerator  can  be  divided  tcithout  remainder. 

But  when  a  remainder  would  be  left,  it  is  best  to  di- 
vide,  by  midtiiilying  the  denominator.  The  answer  is  of 
the  same  value  either  way,  though  the  name  is  different. 

For  example  ;  in  dividing  |  by  2,  we  are  to  find  how 
many  times  2  is  contained  in  |.  Divide  by  midtiplying 
the  denominator  hy  2,  and  we  find  that  it  is  contained  not 
once,  but  -^-^  of  once.  By  dividing  the  numerator  b)"^  2,  we 
find  also  that  it  is  contained  not  once,  but  |  of  once.  Now 
I  and  j'L  is  the  same  value,  by  a  different  name.  For  if 
a  thing  is  divided  into  eighteen  parts,  and  we  take  four  of 


140 


ARITHMETIC.       SECOND  PART. 


them,  we  have  the  same  value  as  if  it  were  divided  into 
nine  parts  and  we  tookfjfjo  of  them. 

Divide  the  following  by  both  methods,  and  explain  them 
as  above. 


Divide 


f     by 


1  7 
16 
5  0 

S-0 

9  0 
114 

250 


3 

4 

5 

8 

10 

12 


Divide 


JL 

by 

2 

I  2 

"J 

1  8 
20 

6 

ai 

7 

£4 

27 

9 

6  5 

_7JL 

10  0 

11 

AV 

9 

Rule  for  Division  where  the  Dividend  is  a 

Fraction. 
Divide  the  numerator  of  the  Fraction  by  the  Divisor,  or, 
[if  this  would  leave  a  remainder,)  multiply  the  denominator 
by  the  Divisor. 

Examples  for  the  Slate. 


In  the  following  examples,  divide  the  numerator  by  the 
divisor. 


Divide       ||     by 


4 

5 

8 

10 

12 


Divide       i||    by 


161 
3  C  6 
.'L6_ 
36  9 
ISO. 
55  9 


11 

16 

7 

75 


In  the  fi.llowing  examples  multiply  the  denominator  by 
the  divisor. 


Divtde 


bv 


4 
6 

8 
12 
24 


Divide         |     by 


5 

7 

9 

12 

61 


30 
4  2 

by 

3 

3  a 

I  2  0 

(( 

8 

_6_4_ 
9  I  2 

<( 

8 

49 

C£ 

T 

3  2  0 

28 

ii 

4 

54 

DIVISION  OF  VULGAR  FRACTIONS.  141 

In  the  following  examples  divide  the  numerator  by  the 
divisor. 

Divide         U     by     3     Divide         ff     by       6 

"                       _3_6_  "         12 

4  6  0  *** 

(C                            8.i  £<              Q 

9  2 

«                          4.2  ii             fi 

((  36  li  Q 

SO  " 

Examples  for  Mental  Exercises. 

1.  If  you  have  f  of  an  orange,  and  wish  to  divide  it 
equally  between  2  children,  what  part  do  you  give  each  ? 

2.  If  you  have  f  of  a  load  of  hay,  and  divide  it  equally 
among  6  horses,  how  much  do  you  give  each  ? 

3.  If  you  have  /_  of  a  yard  of  muslin,  and  divide  it  into 
3  equal  parts,  what  part  of  a  yard  is  each  part  ? 

4.  If  you  have  |f  of  an  ounce  of  musk,  and  divide  it 
into  12  equal  portions,  what  part  of  an  ounce  is  each  por- 
tion ? 

5.  If  you  divide  !§  of  a  dollar  into  4  equal  parts,  what 
part  of  a  dollar  will  each  part  be  ? 

6.  If  a  man  owns  if  of  a  cargo,  and  divides  it  equally 
among  4  sons,  how  much  does  he  give  each  ? 


Division  of  one  Fraction  by  another. 

When  one  fraction  is  to  be  divided  by  another,  the 
same  principle  is  employed,  as  when  whole  numbers  are 
divided  by  a  fraction. 

For  example,  if  the  whole  number  12  is  to  be  divided 
by  ^,  we  first  muUiply  by  the  denominator  4,  to  find  how 
often  one  fourth  is  contained  in  12,  and  then  divide  by  3, 
to  find  how  often  three  fourths  are  contained  in  it. 

In  like  manner,  if  we  wish  to  find  how  many  times,  or 
•parts  of  a  time,  |  is  contained  in  f-^,  we  first  find  how  often 
one  fourth  is  contained  jn  it,  by  reasoning  thus  : 

One  unit  would  be  contained  in  f_,  two  twelfths  of 
one  time. 

One  fourth  would  be  contained  four  times  as  often,  or 
Jj  of  one  time. 


142  ARITHMETIC.       SECOND    PART. 

We  thus  find  how  often  one  fourth  is  contained  in  -^^,  by 
mukiplying  it  by  4,  thus  : 

12       '^      ^     1  2  • 

But  three  fourths  would  be  contained  only  one  third  as 
often,  and  we  find  a  third  of  -^^  ^V  multiplying  its  deno- 
minator by  3.  For  when  we  wish  to  divide  a  fraction 
by  3,  we  multiply  its  denominator,  and  thus  make  the 
parts  represented  by  the  denominator,  three  times  smaller, 
thus  : 

T^    ^^    "^     36' 

Here  the  twelfths  are  changed  to  thirty-sixths  ;  and  a 
thirty-sixth  is  a  third  of  one  twelfth. 

Again  let  |  be  divided  by  |. 

It  will  be  found  by  examining  the  foregoing  process, 
that  in  dividing  one  fraction  by  another,  the  fraction  which 
is  the  dividend  has  its  numerator  midtiplied  by  the  denom- 
inator of  the  divisor,  and  its  denominator  rnultiplied  by  the 
numerator  of  the  divisor. 

Let  another  example  be  taken  and  observe  thus. 

Let  I  be  divided  by  |. 

I  if  divided  by  one  unit  would  contain  it  not  once  but  | 
of  once.  But  if  divided  by  one  sixth  it  would  contain  it 
6  times  as  often  or  6  times  f  which  is  '^- . 

Here  the  numerator  of  the  dividend  (|)  has  been  multi- 
plied by  the  denominator  of  the  divisor  (^),  and  we  have 
thus  found  how  often  one  sixth  is  contained. 

Four  sixths  would  be  contained  only  one  fourth  as 
often,  and  we  therefore  divide  '^p  by  4  by  multiplying  its 
denominator  and  the  answer  is  if,  and  here  the  denomina- 
tor of  the  dividend  has  been  multiplied  by  the  numerator 
of  the  divisor  (|), 

We  therefore  multiplied  the  numerator  of  the  dividend 
by  the  denominator  of  the  divisor  to  find  how  often  one  sixth 
was  contained,  and  multiplied  the  denominator  of  the  divi- 
dend by  the  numerator  of  the  divisor  to  find  how  oftenybwr 
sixths  were  contained. 

Let  the  following  be  performed  and  explained  as 
above. 

Divide     |     by     |        Divide     f     by     f 


DIVISION    OP   VULGAR   FRACTIONS.  143 

This  process  corresponds  with  that  used  in  dividing  a 
ivliole  number  by  a  fraction. 

For  if  we  divide  12  by  |  we  first  multiply  it  by  4  to  find 
how  many  one  fourths  there  are  in  12,  and  then  divide  the 
answer  by  3  to  find  how  many  three  fuurths  there  are. 

So  in  dividing  |  by  |  we  first  multiply  it  by  4  to  find 
how  many  times  onejourth  is  contained  thus  (|),  and  then 
divide  it  by  3  to  find  how  many  times  three  fourths  are  con- 
tained thus,  {x%)' 

ExAMl'LKS. 

Divide     |     by     «         Divide     |     by     fj 


We  invert  a  fraction  when  we  exchmge  the  places  of 
the  numerator  and  the  denominator. 

Thus  i  inverted  is  \,  and  |  inverted  is  f  and  ^a  in- 
verted is  f  I  &c. 

Now  it  appears,  as  above,  that  if  we  wish  to  divide  ' 
by  I  we  are  to  niulti[)ly  its  numerator  (3)  by  the  denomi- 
nator (6)  and  its  denominator  (4)  by  the  numerator  (2). 
This  is  more  easily  done,  it'  wo  incert  the  divisor  f,  thus  #. 

When  the  divisor  is  thus  inverted  we  can  multiply  the 
numerators  together  for  a  new  numerator  and  the  denomi- 
nators for  a  new  denominator  and  the  process  is  the  same. 

Thus  let  us  divide  a  by  |. 

Invertmg  the  divisor  |  the  two  fractions  v.  ould  stand  to- 
gether thus  I  f .  We  now  multiply  the  numerators  and 
denominators  together  and  the  answer  is  ||  and  it  is  the 
same  process,  as  if  we  had  not  inverted  the  divisor,  but 
multiplied  the  numerator  of  the  dividend  by  the  denomi- 
nator of  the  divisor  and  its  denominator  by  the  numerator 
of  the  divisor. 

This  method  therefore  is  given  as  the  easiest  rule,  but 
it  must  be  remembered  that  in  this  process  we  always  mul- 
tiply the  dividend  by  the  denominator  of  the  divisor  and  di- 
vide it  by  the  numerator,  as  we  do  in  case  of  whole 
numbers. 


144 


ARITHMETIC.       SECOND    PART. 


Common  rule  for  dividing  one  fraction  by  another. 

Invert  the  Divisor,  and  then  multiply  the  numerators  and 
denominators  together. 

Examples  for  the  slate. 

Divide  If  by  ~'^. 

Invert  tlie  divisor  and  the  fractions  stand  thus  |f  L^. 

Multiply  them  together,  and  the  answer  is  f  i|. 


87 
9n 

by 

5.4 

75 

Divide 

1  s 

T9" 

by 

72 
8  5 

32 

(( 

5  6 

(C 

ll  4 

C( 

14 

49 

3  9 

1T9 

TT 

IS. 

cc 

32 

(C 

5  6 

(C 

_9JL 

56 

2  1 

412 

506 

6_5_ 

u 

34X 

:c 

J.  6. 

ll 

9_3_ 

138 

30  2 

49 

1  02 

DECIMAL  DIVISION. 

In  order  to  understand  the  process  of  Decimal  Division, 
it  is  needful  to  recollect  the  method  of  dividing  and  multi- 
plying, by  ciphers  and  a  separatrix. 

If  we  wish  to  multiply  a  number  by  a  sum  composed  of 
1  with  ciphers  added  to  it,  we  add  as  many  ciphers  to  the 
multiplicand,  as  there  are  ciphers  in  the  multiplier.  Thus 
if  we  wish  to  multiply  64  by  10,  we  do  it  by  adding  one 
cipher,  040.  If  we  are  to  multiply  by  100,  we  add  two 
ciphers  thus,  6400,  die. 


Multiply     3  by 
"  19  " 


Examples. 

100  Multiply  46  by     100 
1000       "  2    "  100000 


If  we  wish  to  multiply  a  decimal  by  any  number  com- 
posed of  1  with  ciphers  annexed,  we  can  do  it  by  removing 
the  separatrix  as  many  orders  to  the  right,  as  there  are  ci- 
phers in  the  midtiplier. 

Thus  if  ,2694  is  to  be  multiplied  by  10,  we  do  it  thus  ; 
2,694.     If  it  is  to  be  multiplied  by   100,  we  do  it  thus ; 


DECIMAL    DIVISION. 


145 


26,94.  If  it  is  to  be  multiplied  by  1000  we  do  it  thus  ; 
269,4.  But  to  multiply  by  a  million,  we  must  add  ci- 
phers also,  ill  order  to  be  able  to  move  the  separatrix  as 
far  as  required,  thus  ;  209400,. 


Examples. 


Multiply  2,64 
"  36,9468 
«  3,2 


by 


10 

100 
"  1000 


Multiply      6,4 

1,643 

3,2 


by  10000 
"  10 
"  1000000 


The  same  method  can  be  employed  in  dividing  deci- 
mals, by  any  number  composed  of  1  and  ciphers  an- 
nexed. 

The  rule  is  this.  Remove  the  separatrix  as  many  or- 
ders to  the  left,  as  there  are  ciphers  in  the  divisor. 

Thus  if  we  wish  to  divide  23,4  by  10  we  do  it  thus ; 
2,34. 

If  we  wish  to  divide  it  by  100  we  do  it  thus,  234.  But 
if  we  wish  to  divide  it  by  a  thousand  it  is  necessary  to  prc- 
fix  a  cipher  thus  ,0234.  If  we  divide  it  by  10,000  we  do 
it  thus  ,00234. 


Examples. 

Divide       2,4 

by 

100 

Divide     24,3 

by           10 

2,46 

10 

246,9 

100 

3,2 

1000 

2,3 

"  100000 

"            2,4 

10 

34,26 

1000 

Multiply    2,4 

10 

Mult'y.  34,26 

1000 

Divide  328,94 

100 

Divide       3,2 

"     10000 

Mult'y.  326,94 

100 

Multiply    3,2 

"     10000 

It  is  needful  to  understand  that  a  mixed  decimal,  can 
be  changed  to  an  improper  decimal  fraction. 

For  example,  if  we  change  3,20  to  an  improper  decimal 
fraction,  it  becomes  320  hundredths  (f|^),  which  is  an 
improper  fraction,  because  its  numerator  is  larger  than 
the  denominator. 

But  we  cannot  express  the  denominator  of  320  hun- 
dredths, by  a  separatrix  in  the  usual  manner,  for  the  rule 
requires  the  separatrix  to  stand,  so  that  there  will  be  as 
13 


146  ARITHMETIC.       SECOND  PART. 

many  figures  at  the  right  of  it,  as  there  are  ciphers  in  the 
denominator. 

If  then  we  attempt  to  write  320  hundredths  in  this  way, 
it  will  stand  thus  3,20,  which  is  then  a  mixed  decimal,  and 
must  be  read  three  units  and  20  hundredths.  If  it  is  writ- 
ten  thus,  3.|^,  it  is  then  a  vulgar  and  not  a  decimal  frac- 
tion. 

But  it  is  convenient  in  explaining  several  processes  in 
fractions,  to  have  a  method  for  expressing  improper  deci- 
mal jractions,  without  writing  their  denominator.  The  fol- 
lowing method  therefore  will  be  used. 

Let  the  inverted  separatrix  be  used  to  express  an  im- 
proper decimal  fraction.  Thus  let  the  mixed  decimal  2,4 
which  is  read  two  and  four  tenths,  be  changed  to  an  im- 
proper decimal  thus,  2'4  which  may  be  read  twenty-jour 
ttnths. 

The  denominator  of  an  improper  decimal,  (like  that  of 
other  decimals)  is  always  1  and  as  many  ciphers  as  there 
are  figures  at  the  right  of  the  separatrix.  It  is  known  to 
be  an  improper  decimal,  simply  by  liaving  its  separatrix 
inverted. 

Thus  24*69  is  read,  two  thousand  four  hundred  and  six- 
ty-nine hundredths.  239'6  is  read,  two  thousand  three 
hundred  and  ninety-six  tenths,  &c. 

Examples. 

Change  the  following  mixed  decimals  to  improper  deci- 
mals, and  read  them. 

246,3  24,96  32,1 

326,842  3,6496  49,2643 

8,4692  368,491  26,3496 

Rule  for  writing  an  Improper  Decimal. 

Write  as  if  the  numerator  were  whole  numbers,  and  place 
an  inverted  separatrix,  so  that  there  ivill  be  as  mani/ figures 
at  the  right,  as  there  are  ciphers  in  the  denominator. 

Write  the  following  improper  decimals. 

Three  hundred  and  six  tenths. 

Four  thousand  and  nine  hundredths. 


DECIMAL    DIVISION.  147 

Two  hundred  and  forty-six  thousand,  four  hundred  and 
six  tenths. 

Three  milHons,  five  hundred  and  forty-nine  tens  of  thou- 
sandths. 

Two  hundred  and  sixty-four  thousand,  five  hundred  and 
six  thousandths. 

Five  luindred  and  ninety-six  tenths. 

Decimal  Division  when  the  Divisor  is  a  whole  number. 

The  rules  for  Decimal  Division   are   constructed  upon 
this  principle,  that  any  quotient  figure  must  always  be  put 
N|  in  the  same  order  as  the  lowest  order  of  that  part  of  the 
dividend  taken. 

Thus  if  we  divide  ,2.5  (or  two  tenths',  five  hundredths,) 
by  5,  the  quotient  figure  must  be  put  in  the  hundredth  or- 
der, thus,  (,05)  because  the  lowest  order  of  the  dividend  is 
hundredths. 

Again,  if  ,250  is  divided  by  50,  the  quotient  figure  must 
be  5  thousandths,  (,005)  for  the  same  reason. 
'        Let  us  then  divide  ,256  by  2.     VVc  proceed  exactly  as 
f    in  the  Short  Division  of  whole  numbers,  except  in  the  use 
of  a  separalrix. 

Let  the  pupil  proceed  thus  : 

2),256 
,128 

2  tenths  d-vided  by  2,  gives  1  as  quotient,  which   is  1 
tenth,  and  is   set  under  that  order  with  a  separatrix  before 
it.     5  hundredths  divided  by  2,  gives  2  as  quotient,  which 
is  2  hundredths,  and  is  set  under  that  order. 
'  1  hundredth  remains,   which  is  changed  to  thousandths, 

and  added  to  the  6,  making  Iti  thousandths. 

This,  divided  by  2,  gives  8  thousandths  as  quotient,  which 
is  placed  in  that  order. 

If  the  divisor  is  a  whole  number,  and  has  several  or- 
ders in  it,  we  proceed  as  in  Long  Division,  except  we  use 
a  separatrix,  to  keep  the  figures  in  their  proper  order. 
Thu3  if  we  divide  15,12  by  36,  we  proceed  thus  : 


148  ARITHMETIC.       SECOND  PART 

36)15,12(,42 
14,4 


,72 

,72 

,00 

We  first  take  the  15,1  and  divide  it,  remembeiing  that 
the  quotient  figure  is  to  be  of  the  same  order  as  the  lowest 
order  in  the  part  of  the  dividend  taken,  of  course  the  quo- 
tient 4  is  4  tenths  (,4)  and  must  be  written  thus  in  the 
quotient. 

We  now  subtract  36  times  ,4  which  is  14,4.  (See 
rule  for  Decimal  Multiplication  page  108)  from  the  part 
of  the  dividend  taken  and  7  terdhs  (.7)  remain. 

To  this  biing  down  the  2  hundredtlis.  Divide,  and  the 
quotient  figure  is  2  hundredths  which  must  be  set  in  that 
order  in  the  quotient. 

Subtract  36  times  ,02  (or  ,72)  from  the  dividend  and 
nothing  remains.  , 

Let  the  following  sums  be  performed  and  explained  as 
above. 


Divide       76,8     by     24 
94,6      "      43 


Divide       37,8     by     21 
85,8      "     20 


Sometimes   ciphers  must  be  prefixed  to  the  first  quotient 
figure,  to  make  it  stand  in  its  proper  order. 

For  example,  let  ,1512  be  divided  by  36,  and  we  pro- 
ceed thus, 

36),1512(,0042 
,144 

,0072 
,0072 

0000 


.    DECIMAL    DIVISION.  149 

We  take  ,151  first,  which  is  151  thousandths  (for  the 
denominator  of  any  decimal  is  always  of  the  same  order 
as  the  lowest  order  taken). 

This  divided  by  36  gives  4  as  quotient.  This  4  is  4 
thousandths,  because  the  lowest  order  in  the  part  of  the 
dividend  taken  is  thousandths.  Therefore  when  it  is  put 
in  the  quotient  it  must  have  two  ciphers  and  a  separalrix 
prefixed  thus  ,004. 

We  now  subtract  from  the  dividend  36  times,  ,004  or 
,144.     (See  rule  for  Decimal  Multiplication.) 

It  is  desirable  in  such  cases  to  place  ciphers  and  a  sep- 
aratrix  in  the  remainders,  to  make  them  stand  in  their 
proper  orders. 

To  the  remainder  (,007)  bring  down  the  2  tens  of 
thousandths  making  72  tens  of  thousandths. 

This  divided  by  36  gives  "2  tens  of  thousandths  as  quo- 
tient which  is  set  in  that  order.  36  times  2  tens  of  thou- 
sandths  (or  ,0072)  being  subtracted,  nothing  remains. 


Sometimes  we  must  add  ciphers  to  the  ditndend  before  toe 
can  begin  to  divide. 

For  example,  let  ,369  be  divided  by  469,  and  we  pro- 
ceed thus, 

469),3690(,00078 
,3283 


,04070 
,03752 

,00318 


We  find  that  ,369  cannot  be  divided  by  469,  so  we  add 
a  cipher  to  it,  making  it  3690  tens  of  thousandths. 

This  divided  by  469  gives  7  as  quotient,  which  is  7  tens 
of  thousandths,  (,0007)  because  the  lowest  order  of  the 
dividend  is  of  that  order. 

We  now  subtract  469  times  ,0007  (which  is  ,3283)  from 
the  dividend,  and  ,0407  remain. 
13* 


150  ARITHMETIC.       SECOND    PART. 

To  this  remainder  we  add  a  cipher,  and  change  it  fron» 
407  tens  of  thousandths  to  4070  hundreds  of  thousandths. 

This  divided  by  469  gives  8  as  quotient,  which  is  8  huu' 
dreds  of  thousandths,  because  the  lowest  order  in  the  div- 
idend is  hundreds  of  thousandths. 

We  now  subtract  469  times  8  hundreds  of  thousandths 
(or  ,03752)  from  the  dividend  and  ,00318  remain. 

We  could  continue  dividing,  by  adding  ciphers  to  the 
remainders,  but  it  is  needless.  Instead  of  this  we  can  set 
the  divisor  under  the  remainder  as  in  common  division, 
thus  ^J-^ 

'""^4  6  3 

It  is  not  needful  to  retain  the  separatrix  and  ciphers 
when  thus  writing  a  remainder,  because  when  put  in  the 
quotient,  it  is  not  considered  as  the  ^^^  part  oi'  a  whole 
number,  but  as  a  part  of  the  lowest  order  in  the  decimal, 
by  which  it  is  placed. 

Thus  when  this  is  put  with  the  above  quotient,  we  read 
the  answer  thus  78  hundreds  of  thousandths,  and  |i|  of 
another  hundred  oj  thousandth. 

Let  the  following  sums  be  performed  and  explained  as 
above. 

Divide 


3,694 

bv     84 

Divide     428G9 

by 

95 

,36946 

''   841 

"      3,69428 

li 

49 

3,26 

"■   589 

,269 

li 

482 

32,4 

"   386 

481,4 

(( 

81 

364,6 

"     99 

28,1 

(( 

15 

Decimal  Division  when  the  Divisor  is  a  Decimal. 

Vilhen  the  divisor  is  a  decimal,  we  proceed  as  in  divi- 
ding  by  a  Vulgar  Fraction,  viz. 

We  multiply  by  the  denominator,  and  divide  ly  the  nu- 
merator. 

Thus  if  we  are  to  divide  24  by  ,4,  we  are  to  find  how  ma- 
ny 4  tenths  there  are  in  24. 

We  first  multiply  24  by  the  denominator  10,  to  find  how 
many  one  tenths  there  are,  and  then  divide  by  the  numer- 
ator 4,  to  find  how  many  4  tenths  there  are.  24  is  multi- 
plied by  tea,  thus  ;  24'0,  and  has  the  inverted  separatrix, 
to  show  that  it  is  not  240  wlmle  nu?nbers,  but  tenths. 

We  now  have  found  that  in  24  there  are  240  owe  tenths, 


DECIMAL    DIVISION. 


151 


we  now  divide  by  4,  to  find  how  many  4  tenths  there  are. 
The  answer  is  60,  which  according  to  the  rule,  must  be  of 
the  same  order  as  the  lowest  order  in  the  dividend,  or  60 
tenths,  and  must  be  shown  by  the  inverted  separatrix  thus 
(t)'0.)  This  may  be  clianged  to  whole  numbers  by  revert- 
ing the  separatrix  thus  (0,0.) 

When  the  dividend  is  a  decimal,  we  can  multiply  by  re- 
moving the  separatrix. 

Thus  let  8,64  be  divided  by  ,36. 

Here  we  are  to  multiply  by  100,  to  find  how  many  one 
hundredths  there  are  in  the  dividend,  and  then  divide  by 
36  to  find  how  many  36  hundredths  there  are. 

We  multiply  by  100,  by  removing  the  separatrix  two 
orders  toward  the  right,  and  then  dividing  by  36,  we  have 
24  as  answer,  which  is  24  units,  because  the  dividend  is 
units,  as  appears  below. 

36)804,(24 
72 


144 
144 

000 


If  the  divisor  is  a  mixed  decimal,  we  change  it  to  an  im- 
proper decimal,  and  then  proceed  !is  betbre,  multiplying 
by  the  (lenominator  and  divide  by  the  numerator. 

Thus  let  10,58  be  divided  by  4,6, 

We  first  change  the  divisor  into  an  improper  decimal 
thus,  4'6  {^iitenths.) 

We  now  are  to  multiply  the  10,.58  by  10,  to  find  how 
many  one  tenths  there  are,  and  then  divide  by  46,  to  find 
how  many  40  tenths  there  are. 

We  multiply  by  10  by  removing  the  separatrix  thus, 
105,8,  and  p-roceed  as  follows. 


152  ARITHMETIC.       SECOND   PART. 

46)105,8(2,3 
92 


000 


Here  we  divide  105  units  by  46,  and  the  quotient  fig- 
ure is  2  units. 

We  then  subtract  46  times  2  units  from  the  dividend, 
and  13  units  remain.  To  this  bring  down  the  8  tenths. 
This  is  divided  as  if  whole  numbers,  but  the  quotient  3  is 
3  tenths,  because  the  lowest  order  in  the  dividend  is  tenths. 
It  is  set  in  the  quotient  with  the  separatrix  before  it,  and 
then  46  times  ,.3  (or  13,8)  is  taken  from  the  dividend,  and 
nothing  remains. 

Let  the  following  sums  be  performed,  and  explained  as 
above. 


46,4 

by         3,6 

Divide     891,6   by 

,2 

,431 

2,41 

"         8,964     " 

8,6 

4,56 

3,64 

"          89,96     " 

4,861 

464,92 

"   3,2649 

"          8,641     " 

,4169 

Divide 


The  following  then  is  the  rule  for  Decimal  Division. 
Rule  for  Decimal  Division. 

IJ  the  divisor  is  a  whole  number,  divide  as  in  common  di' 
vision,  plachig  each  quotient  Jigure  in  the  same  order  as  the 
lowest  order  of  the  dividend  taken. 

If  the  divisor  is  a  decimal,  multiply  by  the  denominator, 
and  divide  hy  the  numerator,  placing  each  quotient  figure  in 
the  same  order  as  the  lowest  order  of  the  dividend  taken. 

If  the  divisor  is  a  mixed  decimal,  change  it  to  an  improper 
decimal,  and  then  proceed  to  midiiply  by  the  denominator 
and  divide  hy  the  numerator. 

N.  D .  The  rule  for  multiplying  and  dividing  Federal 
Money,  is  the  same  as  for  Decimals. 


decimal  division.  153 

Examples. 

How  many  times  is  $2,04  contained  in  $9,40  ? 
Divide     $2,04       by     $,84 
02  "         8,41 

"  2,41        «         19,24 

324,07   «         64,81 
"  20,46      "         ,49 

As  it  is  found  to  be  invariably  the  case  tljat  the  decimal 
orders  in  the  divisor  and  quotient  always  equal  those  of  the 
dividend,  the  common  rule  for  decimal  division,  is  formed 
on  that  principle,  and  may  now  be  used. 

Common  Rule  for  Decimal  Division. 

Divide  as  in  whole  numbers.  Point  off  in  the  quotient 
enough  decimals  to  make  the  decimal  orders  of  the  divisor 
and  quotient  together  equal  to  those  of  the  dividend,  counting 
every  cipher  annexed  to  the  dividend,  or  to  any  remainder,  as 
a  decimal  order  of  the  dividend.  If  there  are  not  enough 
figures  in  the  quotient  prefix  ciphers. 

In  pointing  off  by  the  above  rule,  let  the  teacher  ask 
these  questions. 

How  many  decimals  in  the  dividend  ?  How  many  in 
the  divisor  ?  How  many  must  be  pointed  off  in  the  quo- 
tient, to  make  as  many  in  the  divisor  and  quotient,  as  there 
are  in  the  dividend  ? 

Examples. 

At  $,75  per  bushel,  how  many  bushels  of  oats  can  be 
bought  for  814,23? 

How  much  butter  at  16  cents  a  pound,  can  be  bought  for 
$20? 

A  half  cent  can  be  written  thus,  $,005  (for  5  mills  is 
half  a  cent,  or  5  thousandths  of  a  dollar.) 

A  quarter  o^ a.  cent  can  be  written  thus,  $,0025  (for  \  of 
a  cent  is  25  tens  of  thousandths  of  a  dollar.) 

At  \2\  cents  per  hour,  in  how  much  time  will  a  man 
earn  <S46  ? 


154  ARITHMETIC.       SECOND  PART. 

At  6J^  cents  per  pint,  how  much  molasses  may  be  bought 
for  $2  ? 

At  $,06  an  ounce,  how  much  camphor  can  be  bought 
tor  $3  ? 

At  $,l2i  a  bushel,  how  much  coal  could  be  bought  for 
85?  '    ■    ^ 

Divide  ,032  by  ,005. 

Exercises  in  Decimal  Multiplication  and  Division. 

Multiply  ,25  by  ,003.     Divide  ,25  by  ,003. 

Multiply  3,4  by  2,68.     Divide  3,4  by  2,68. 

Multiply  ,005  by  ,005.     Divide  ,004  by  16,4. 

If  you  buy  24  bushels  of  coal,  at  $,09  per  bushel,  what 
does  the  whole  cost? 

If  a  man's  wages  be  fifty  hundredths  of  a  dollar  a  day, 
what  will  it  be  a  month  ? 

What  will  be  the  cost  of  25  thousandths  of  a  cord  of 
wood,  at  $2  a  cord  1 

What  will  be  the  cost  of  twelve  hundredths  of  a  ton  of 
hay,  at  $11  a  ton? 

If  a  man  pays  a  tax  of  two  mills  on  a  dollar,  how  much 
must  he  pay  if  he  is  worth  ^350  ? 

If  a  man  pays  $,06  a  year  for  the  use  of  each  dollar  he 
borrows  of  his  neighbor,  how  much  must  he  pay  in  a  year 
if  he  borrows  264  dollars  ?     How  much  in  two  years  ? 


REDUCTION. 

Reduction  is  changing  units  of  one  order,  to  units  of 
another  order. 

Reduction  Ascending,  is  changing  units  of  a  lower  to  a 
higher  Older. 

Reduction  Descending,  is  changing  units  o(?k  higher  to  a 
lower  order. 

Examples  for  Mkntal  Exercise. 

In  4  gallons  how  many  quarts  ? 

Note.     Let  each  sum  be  stated  thus.     One  gallon  con- 


REDUCTION.  155 

tains  four  qoarts,  and  four  gallons  four  times  as  much.  4 
times  4  is  16. 

In  4  gallons  how  many  pints  ? 

In  8  yds.  3  qrs.  how  many  quarters  ? 

In  8  feet  how  many  inches  ? 

In  4  bushels  how  many  quarts  ? 

In  5  hours  how  many  minutes  ? 

Are  the  above  sums  in  Reduction  Ascending  or  De- 
scending ? 

In  32  quarts  how  many  gallons  ? 

Let  such  sums  be  stated  thus.  One  gallon  contains  4 
quarts.  In  32  quarts  therefore,  there  are  as  many  gal- 
Ions  as  there  are  4's  in  32. 

In  42  pints  how  many  gallons  ? 

In  49  quarters  how  many  yards  ? 

In  50  nails,  how  many  quarters  and  how  many  yards  ? 

In  64  inches  how  many  feet  ? 

In  36  barley  corns  how  many  inches  ? 

In  96  quarts  how  many  bushels  ?  • 

In  120  minutes  how  many  hours  ? 

In  48  feet  how  many  yards  ? 

In  94  inches  how  many  feet  ? 

In  3  yards  how  many  inches  1 

In  4  gallons  how  many  pints  ? 

In  32  quarts  how  many  gallons  ? 

In  80  penny  weights  how  many  ounces  ? 

In  24  ounces  how  many  penny  weights  ? 

In  8  pounds  how  many  shillings  ? 

In  40  shillings  how  many  pence  ? 

In  £2,  9s.  6d.  3  qrs.  how  many  farthings  ? 

In  doing  this  sum  we  proceed  in  the  following  manner : 

£.         s.        d.       qr. 
2    «    9  "    6    «  3 
20 

49  shillings. 

594  pence. 
4 


2379  farthings. 


156  ARITHMETIC.       SECOND    PART. 

We  first  change  the  pounds  to  shillings,  by  multiplying 
by  20,  and  add  the  9  shillings  to  them,  making  49  shil- 
lings. 

We  then  change  the  49  shillings  to  pence,  by  multiply- 
ing by  12,  and  add  the  G  pence  to  them,  making  594 
pence. 

We  then  change  the  594  pence  to  farthings,  by  multi- 
plying by  4,  and  add  the  3  qrs.  and  thus  we  obtain  the  an- 
swer 2379  qrs 

This  is  Reduction  Descending^  because  we  have  chan- 
ged units  of  a  higher  order  to  those  of  a  lower. 

Why  did  we  multiply  by  20,  12,  and  4? 

Let  us  now  reverse  tlie  process,  and  change  2379  far- 
things to  pounds. 

We  proceed  thus : 

£.       s.       d.       qr. 
4)2379(2    "9    "6    "  3. 
•  12)594 

20)49 
2 

We  first  change  the  2379  farthings  to  pence,  by  divi- 
ding by  4,  and  the  answer  is  594  pence,  and  3  farthings 
(or  qr.)  over,  which  is  put  in  the  quotient  with  qr.  over  it. 

We  then  change  the  594  pence  to  shillings,  by  dividing 
by  12,  and  the  answer  is  49  shillings,  and  six  pence  over, 
which  is  put  in  the  quotient  with  d.  written  over. 

We  next  change  the  49  shillings  to  pounds,  by  dividing 
by  20,  and  find  there  is  £2  and  9s.  over,  which  are  both 
put  in  the  quotient  with  their  signs  written  over  them. 

Why  did  we  divide  by  4,  12,  and  20  ? 

Let  the  following  sums  be  performed  and  explained  in 
the  same  way. 

Change  2486  farthings  to  pounds. 

Change  £2  18s.  4d.  2qr.  to  farthings. 

Change  241  shillings  to  pounds. 

Change  249  pence  to  shillings  and  pounds. 

Change  £21  2s.  to  farthings.  i 

Change  361  pounds  to  pence. 

Change  35  shillings  to  pounds. 


REDUCTION.  157 

Rule  for  Reduction. 
To  reduce  from  a  higher  to  a  lower  order. 
Multiply  the  highest  order  by  the  member  required  of  the 
next  lower  order,  to  make  a  unit  of  this  order. .  Add  the 
next  lower  order  to  this  product,  and  midtiphj  it  by  the  num. 
ber  required  of  the  next  lower  order,  to  make  a  unit  of  this 
order,  adding  as  before.     Thus  through  all  the  orders. 

To  reduce  from  a  lower  to  a  higher  order. 
Divide  the  amount  given,  by  the  number  required  to  make 
a  unit  of  the  next  higher  order.  Divide  the  ansicer  in  the 
same  way,  and  continue  thus  till  the  answer  is  in  units  of  the 
order  demanded.  The  remainders  are  of  the  same  order 
as  the  dividend,  and  are  to  be  put  as  a  part  of  the  answer. 

Exercises. 

Bought  a   tankard  of  silver  weighing  5  lb.    3  oz.  for 
which  I  paid  $1,12  an  oz.  how  much  did  it  cost  ? 

Reduce  2  lb.  8oz.  11  pwt.  to  grains. 

In  81b.  93.  43.  2  9.  16grs.  how  many  grains? 

In  11924  grains  how  many  pounds  ? 

What  cost  4  cwt.  3  qrs.  17  lb.  of  sugar,   at  I2i  cents 
per  lb? 

In  436  boxes  of  raisins,  each  containing  24  lbs.  how 
many  cwt.  ? 

In  63469542  drams,  how  many  tons  ? 

In  546  yards  how  many  nails  ? 

In  5486  nails  how  many  yards  ? 

In  118|  yards,  how  many  Ells  Flemish  ? 

How  many  barley  corns  will  reach  round  the  globe,  it 
being  360  degrees  ? 

How  many  miles  in  836954621  barley  corns  ? 

In  18  acres,  3  roods,  12  rods,  how  many  square  feet  ? 

How  many  square  feet  in  16  square  miles  ? 

In  9269546231  square  feet  how  many  square  miles  ? 

In  37  cords  of  wood  how  many  solid  feet  ? 

In  20486  solid  feet  how  many  cords  ? 

In  4  pipes  of  wine  how  many  pints  ? 

In  9120854  pints  how  many  pipes  ? 
14 


158         ARITHMETIC.   SECOND  PART. 

In  464  bushels  how  many  quarts  ? 
In  964693  pints  how  many  bushels  ? 


REDUCTION  OF  FRACTIONS  TO  WHOLE 
NUMBERS. 

1.  In  ten  fifths,  how  many  units  ? 

2.  In  fourteen  sevenths,  how  many  units  ? 

3.  Change  fifteen  fifths  to  units. 

4.  Change  thirteen  fourths  to  units,  and  what  is  the  an- 
swer ? 

5.  Change  eighteen  fourths  to  units,  and  what  is  the  an- 
swer? 

6.  Change  fourteen  sixths  to  units.  * 

It  will  be  perceived,  that  in  answering  these  questions, 
the  pupil  divides  the  numerator  by  the  denominator.  Thus 
in  changing  twelve  fourths  to  units,  the  numerator  twelve, 
is  divided  by  the  denominator  four.  The  above  sums  are 
to  be  performed  mentally  first,  and  the  answers  given,  and 
then  they  are  to  be  written,  thus, 

7.  Change  fourteen  sixths  to  units. 
Ans.   L.4  =  14  _^  6  =  2  f 

Let  the  pupil  be  required  to  perform  all  the  above  sums, 
in  this  manner. 


Rule  for  Reducing  Fractions  to  Whole  Numbers. 

Divide  the  numerator  by  the  denominator ;  write  the  re- 
mainder, if  there  be  any,  over  the  denominator,  and  annex 
the  fraction,  thus  formed,  to  the  quotient. 

Examples. 

1.  Reduce  "^^  to  a  whole  or  mixed  number.     Ans.  9|. 

2.  Reduce  V'     ^i^s.    ^.     V-  Ans.   9f.     y.    Ans. 
151.     iJ).  Ans.  2f. 


REDUCTION   OF    FRACTIONS.  159 

3.  Reduce  'f^.     Ans.  52|.     ^sjs.  Ans.  565.   2>|3s, 
Ans.  2425. 

4.  Reduce  ^'-^•.    6?_78,    sisjas.    915873.   i32s.96s. 

5.  Reduce     out  es_H32  l  ^        700070007.        6003il4002. 

6.  Reduce      71I23_*S499.        49S63j50217.  33222_11136. 
59248  32  1768 


REDUCTION  OF  WHOLE  NUMBERS  TO 
FRACTIONS. 

1.  In  three   units,   how  many  fourths,  and  how  is  the 
answer  expressed  in  figures  1 

2.  How  many  fifths  is  three  units  and  two  fifths,  and 
how  is  the  answer  written  ? 

3.  Reduce  nine  units  to  sixths. 

4.  Reduce  seven  units  and  two  twelfths  to  twelfths. 


Rule  for  reducing  Whole  Numbers  to  Fractions. 

Multiply  the  whole  number,  by  the  denominator  of  the 
fraction  to  which  it  is  to  be  reduced,  and  place  the  product 
over  this  denominator.  If  there  is  with  the  units,  a  fraction 
of  the  same  denominator,  add  the  numerator  of  this  fraction 
to  the  product,  before  placing  it  over  the  denominator. 

Examples. 

1.  How  many  4ths.  in  1  ?     How  many  in  1^  ?    In  1^  ^ 

In  If  ?  .^4  4- 

2.  How  many  5ths.  in  1  ?    In  5  ?    In  11  ?    In  l^  ?    In 

74?  5  a 

"•5    • 

3.  How  many  7ths.  in  7  ?    In  8  ?    In  12  ?    In  7| '    In 

5f  ?  ' 

4.  How  many  12ths.  in  9  -^%  ?  In  7  -rV  ?  In  3  A  '?  In 
5/^?  In  8,',? 

5"  How  many  6ths.  in  3  ?  In  4  ?  In  5  #  ?  In  7  |  ?  In 
8  ?  In  9  i  ?  In  12  ?  ' 


160  ARITHMETIC.       SECOND  PART. 

6.  How  many  27ths.  in  3  ?   In  2  ?    In  5  J^  ?    Ans.  |i. 

fl  4        I_4  4  ^ '' 

27*       2  7" 

7.  How  many  I9ths.  in  15  ?     In  13  j\  1     In  17  jf  ? 

Ans     2_8S       2S_0        34  1 


REDUCTION  OF  VULGAR  TO  DECIMAL 
FRACTIONS. 

Decimal  Fractions  are  generally  used  in  preference  to 
Vulgar,  because  it  is  so  easy  to  multiply  and  divide  by 
their  denontiinators. 

Vulgar  fractions  can  be  changed  to  Decimals  by  a  pro- 
cess which  will  now  be  explained. 

In  this  process,  the  numerator  is  to  be  considered  as 
units  divided  by  the  denominator. 

Thus  f  is  3  units  divided  by  4,  for  |  is  a  fourth  of  3  units. 

We  can  change  these  3  units  to  an  improper  decimal 
thus,  3'0  (30  tenths),  and  then  divide  by  4  ;  remembering 
that  the  quotient  is  of  the  same  order  as  the  dividend. 
4)3'0(,75 
2*8 

,20 
,20 

Thus  the  30  tenths  are  divided  by  4,  and  the  answer  is  7 
tenths,  which  is  placed  in  the  quotient,  with  a  separatrix 
prefixed.  4  times  7  tenths  (or  28  tenths)  are  then  sub- 
tracted, and  the  remainder  is  ,2.  This  in  order  to  divide 
it  by  4,  must  have  a  cipher  annexed,  making  it  20  hund- 
redths.  The  quotient  of  this  is  5  hundredths,  and  no  re- 
mainder. 

(In  performing  this  process,  particular  care  must  be 
taken  in  using  the  separatrix,  both  for  proper  and  improper 
decimals.) 

Let  I  be  reduced  in  the  same  way. 

The  two  units  are  first  changed  to  an  improper  decimal 
thus  : 


REDUCTION  OF  FRACTIONS.  161 

8)2'0(,25 
1'6 

,40 
,40 

00 

We  proceed  thus.  20  tenths  divided  by  8,  is  2  tenths, 
which  is  placed  in  the  quotient.  8  times  ,2,  or  16  tenths 
(1'6)  is  then  subtracted,  and  ,4  remain. 

This  is  changed  to  40  hundredths  (,40)  by  adding  a 
cipher,  and  then  divided  by  8.  The  quotient  is  6  hund. 
redths,  vi'hich  is  put  in  the  quotient  and  there  is  no  re- 
mainder. 

Note.  After  3  or  4  figures  are  put  in  the  quotient,  if 
there  still  continues  to  be  a  remainder,  it  is  not  needful  to 
continue  the  division,  but  merely  to  put  the  sign  of  addition 
in  the  quotient  to  show  that  more  figures  might  be  added. 

Examples. 

Reduce  -^-^  to  a  decimal,  and  explain  as  above. 

Reduce  f  f  |  tt  t 3  t  f  f  ^^^^^  ^^  ^  decimal  of  the 
same  value. 

Let  the  pupil  be  required  to  explain  sums  of  this  kind 
as  directed  above,  until  perfectly  familiar  with  the 
principle. 

When  fractions  of  dollars  and  cents  are  expressed,  their 
decimal  value  is  found  by  the  same  process. 

For  example,  change  \  a  dollar  to  a  decimal. 

Here  the  1  of  the  numerator,  is  one  dollar,  divided  by 
2.  By  adding  a  cipher  to  this  1  and  using  the  inverted 
separatrix,  the  dollar  is  changed  to  10  dimes,  and  when 
this  is  divided  by  2,  the  answer  is  5  ;  which  being  of  the 
same  order  as  the  dividend  is  5  dimes. 

The  answer  is  to  be  written  with  the  sign  of  the  dollar 
before  it,  thus  i0,5. 

Tlie  only  difference  between  the  answer  when  1  is  re- 
duced to  a  decimal,  and  when  \  a  dollar  is  reduced  to  a 
decimal,  is  simply  the  use  of  the  sign  of  a  dollar  ($)  and 
a  cipher  in  the  dollar  order. 


162  ARITHMETIC.       SECOND  PART. 

1.  Reduce  1  to  a  decimal.     Ans.  ,b. 

2.  Reduce  \  a  dollar  to  a  decimal.     Ans.  $0,5. 

3.  Change  |  of  a  dollar  to  a  decimal.     Ans.  $0,125. 

4.  Change  yL  of  a  dollar  to  a  decimal.     Ans.  $0,0625. 
In  this  last  sum  there  must  be  two  ciphers  added  to  the 

numerator,  changing  the  1  dollar  to  cents,  instead  of 
dimes  ;  and  in  this  case  a  cipher  is  put  in  the  order  of 
dimes,  and  the  quotient  (being  of  the  same  order  as  the 
dividend)  is  placed  in  the  order  of  cents. 

5.  Reduce  j  of  a  dollar  to  a  decimal.     Ans.  $0,2. 

6.  Reduce  f  of  a  dollar  to  a  decimal.     Ans.  ^0,025. 

7.  Rp.duce  f'-  of  a  dollar  to  a  decimal.     Ans.  |i'0,1871. 

8.  Reduce  Jg  to  the  decimal  of  a  dollar.     Ans.  $0,01. 


Rule   fok   the    reduction   of   Vulgar   to   Decimal 
fractions. 

Change  the  numerator  to  an  improper  decimal,  by  annex- 
ing ciphers  and  using  an  inverted  separatrix.  Divide  by 
the  denominator,  placing  each  quotient  figure  in  the  same  or- 
der as  the  lowest  order  of  the  part  divided. 


1. 

Reduce  ^i^  to  a  decimal. 

Ans. 

.0016. 

2. 

Reduce  ^|^  to  a  decimal. 

Ans. 

.028. 

3. 

Reduce  if  o  to  a  decimal. 

Ans. 

.05625. 

4. 

Reduce  ^  to  a  decimal. 

Ans. 

.3333333+ 

Note.  We  see  here,  that  we  may  go  on  forever,  and 
the  decimal  will  continue  to  repeat  33,  &lc.  therefore,  the 
sign  of  addition  +  in  such  cases  may  be  added,  as  soon 
as  it  is  found  that  the  same  number  continues  to  recur 
in  the  quotient. 


REDUCTION    OF   FRACTIONS   TO  A  COMMON 
DENOMINATOR. 

Before  explaining  this  process,  it  must  be  remembered 
that  fill  &,c.  or  a  iraction  which  has  the  numerator 
and  denominator  alike,  is  the  same  as  a  unit.  If  therefore 
we  take  a  fourth  of  f  it  is  the  same  as  taking  a  fourth  of 


REDUCTION  OF  FRACTIONS.  163 

one.     If  we  take  a  sixth  of  |  it  is  the  same  as  taking  a 
sixth  of  one. 

If  we  take  |  of  f  it  is  the  same  as  taking  |  of  one. 

Whenever  therefore  we  wish  to  change  one  fraction 
to  another,  \vithout  altering  its  value,  we  suppose  a  iinit  to 
be  cha.nged<o a.  fractio7ialfo7-7n,  and  then  take  such  apart 
of  it,  as  is  expressed  by  the  fraction  to  be  changed. 

For  example,  if  we  wish  to  change  j  to  twelfths,  we 
change  a  unltto  twelfths  and  then  take  i  of  it,  and  we  have 
i  of  If,  which  is  the  same  as  i  of  one. 

If  we  wish  to  change  i  to  eighths,  we  change  a  unit  to 
I  and  then  take  i  of  it,  for  i  of  |  is  the  same  as  4  of  one. 

Change  |  to  twelfths,  thus,  a  unit  is  i|.  One  third 
of  is.  is  -^\.     Two  thirds  is  twice  as  mucli,  or  ■^^.     Then 

t  are  tV-  ' 

Change  |  to  twentieths.  A  unit  is  |^.  One  fifth  of  %^ 
is  ^*„.     Four  fifths  is  four  times  as  much,  or  i«. 

Change  the  following  fractions  and  state  the  process  in 
the  same  way. 

Change  |  to  twenty  fourths. 

Change  |  to  twelfths. 

Reduce  ^  to  twenty  sevenths. 

Reduce  f  to  sixty  fourtlis. 

Reduce  |  to  twenty  fifths. 

Reduce  |  to  twenty  sevenths. 

Reduce  ^  to  thirty  sixths. 

Reduce  |  to  forty  ninths. 

Reduce  -^-^  to  thirty  sixths. 

Reduce  ^  to  sixteenths. 

Reduce   ?-  to  fortieths. 

Reduce  |f  to  thirty  thirds. 

Reduce  |  to  thirty  sixths. 

Reduce  |  and  |  each  to  twelfths. 

Reduce  |  and  y\  each  to  twentieths. 

Reduce  i  |  and  |  each  to  twelfths. 

Reduce  \  y-^  and  -^-^  each  to  fortieths. 

Reduce  \  |  jir  and  ^^^  each  to  sixty  fourths. 

Reduce  |  |  -^^  and  -if-  each  to  forty  eighths. 

In  the  above  examples  it  is  seen  that  when  several  frac- 
tions are  to  he  reduced  to  a  common  denominator,  a  V7iit  is 
changed  first  to  a  fractional  form  with  the  required  deno- 


164  ARITHMETIC.       SECOND   PART. 

minator.  Then  it  is  divided  by  the  denominator  of  each 
fraction,  to  obtain  one  part,  and  multiplied  by  the  numera- 
tor, to  obtain  the  required  number  of  parts. 

Thus  changing  |  and  f  each  to  twelfths,  we  first  change 
a  unit  to  a  fraction  with  the  required  denominator  12  ; 
thus,  if.  We  then  divide  it  by  the  denominator  of  |,  to 
obtain  one  fourth,  and  multiply  the  answer  by  3,  to  obtain 
three  fourths.  In  like  manner  with  the  f .  We  divide  || 
by  the  denominator  6,  to  obtain  owe  sixth,  and  multiply  by 
the  numerator  to  obtain  two  sixths. 

In  changing  fractions  to  common  denominators  then,  the 
unit  must  be  changed  to  that  fractional  form  which  will 
enable  us  to  divide  it  by  all  the  denominators  of  the  frac- 
tions  (which  are  to  be  reduced)  without  remainder. 

Thus  if  we  wish  to  reduce  \  and  f  to  a  common  denom- 
iaator,  we  cannot  reduce  them  to  twelfths,  because  ||  can- 
not be  divided  by  either  the  denominator  5,  or  7,  without 
remainder.  We  must  therefore  seek  a  number  that  can 
be  thus  divided,  both  by  7  and  5.  35  is  such  a  number. 
We  now  take  |  of  ||  and  f  of  ||  and  the  two  fractions 
are  then  reduced  to  a  common  denominator. 


One  mode  of  reducing  fractions  to  a  common  deno- 

MINATOR. 

Change  a  unit  to  a  fraction  whose  denominator  can  be 
divided  by  all  the  denominators  of  the  fractions  to  be  redu'. 
ced,  without  remainder.  Divide  this  fraction  by  the  deno- 
minator of  each  fraction  to  obtain  one  part,  and  multiply  by 
the  numerator  to  obtain  the  required  number  of  parts. 

FURTHER  EXAMPLES  FOR  3IENTAL  EXERCISE. 

Reduce  |  |  and  |  to  a  common  denominator. 
Let  the  unit  be  reduced  to  |f . 

Reduce  |  i  |  to  a  common  denominator.  Let  the  unit 
be  reduced  to  ||. 

Reduce  f  f  |  and  /g  to  a  common  denominator. 

Reduce  1 1  f  to  a  common  denominator. 

Reduce  J  |  ^  to  a  common  denominator. 

Reduce  rk  f  I  4  2  ^^  ^  common  denominator. 

But  there  is  another  method  of  reducing  fractions  to  a 


REDUCTION  OP  FRACTIONS.         165 

common  denominator  which  is  more  convenient  for  opera- 
tions on  the  slate.  When  a  fraction  has  both  its  terms 
(that  is  its  numerator  and  denominator)  multiphed  by  the 
same  number,  its  value  remains  the  same. 

For  example  ;  multiply  both  the  numerator  and  deno- 
minator of  I  by  4,  and  it  becomes  ^^.  But  §  and  ^'^  are 
the  same  value,  with  different  names. 

The  effect,  then,  of  multiplying  both  terms  of  a  fraction 
by  the  same  number  is  to  change  their  7iame,  but  not  their 
value. 

If  therefore  we  have  two  fractions,  and  wish  to  change 
them  so  as  to  have  both  their  denominators  alike,  we  can 
do  it  by  multiplication. 

For  example  ; 

Let  I  and  ^  be  changed,  so  as  to  have  the  same  deno- 
minator. This  can  be  done  by  multiplying  both  terms  of 
the  I  by  9,  and  of  |  by  3.  The  answers  are  ^|  and  ^^, 
and  the  value  of  both  fractions  is  unaltered. 

In  this  case  both  terms  of  each  fraction  were  multiplied 
bjf  the  denominator  of  the  other  fraction. 

Let  the  following  fractions  be  reduced  to  a  common  de- 
nominator  in  the  same  way. 

I.  Reduce  |  and  f  to  a  common  denominator.   Multiply 
'the  I  by  the  denominator  7,  and  the  f  by  the  denomina- 
tor  5. 
-  2.  Reduce  f  and  |^  to  a  common  denominator. 

3.  Reduce  |  and  |  to  a  common  denominator. 

4.  Reduce  'y\  and  ^  to  a  common  denominator. 

The  same  course  can  be  pursued,  where  there  are  seV' 
eral  fractions,  to  be  reduced  to  a  common  denominator. 

Thus  if  ^  f  and  |  are  to  be  reduced  to  a  common  deno- 
minator, we  can  multiply  both  terms  of  the  ^  first  by  the 
denominator  3,  and  then  multiply  both  terms  of  the  answer 
by  the  denominator  4,  and  it  becomes  ||  and  its  value  re- 
mains unaltered.  For  a  and  i|  have  the  same  value  with 
a  different  name. 

Then  we  can  multiply  both  terms  of  the  |  first  by  the 
denominator  2,  and  then  by  the  denominator  4,  and  it  be- 
comes i|  and  its  value  remains  unaltered. 

Then  |  may  be  multiplied,  first  by  the  deneminator  2, 


166  ARITHMETIC       SECOND  PART. 

and  then  by  the  denominator  3,  and  it  becomes  i|  and  its 
value  is  unaltered. 

The  three  fractions  J  |  and  a  are  thus  changed  to  if 
^f  and  H  which  have  a  common  denominator,  and  yet 
their  value  is  unaltered. 

But  instead  of  multiplying  each  fraction,  by  each  sepa- 
rate denominator,  it  is  a  shorter  way  to  multiply  by  the 
product  of  these  denominators. 

Thus  in  the  above  example,  instead  of  multiplying  the 
i,  first  by  3,  and  then  the  answer  by  4,  it  is  shorter  to  mul- 
tiply by  12  (the  product  of  3  and  4),  and  the  answer  will 
be  the  same. 

In  like  manner,  if  we  were  to  reduce  |  f  and  ^  to  a 
common  denominator,  we  should  multiply  both  terms  of 
each  fraction  by  the  denominators  of  all  the  other  frac 
tions.  But  instead  of  each  denominator  separately,  as 
multiplier,  we  can  take  the  product  of  them  for  the  mul- 
tiplier. 

Reduce  |  f  and  ^  to  a  common  denominator. 

Here  both  terms  of  the  |  are  first  multiplied  by  the  pro- 
duct of  the  other  two  denominators  (which  is  12).  Then 
both  terms  of  £  are  multiplied  in  the  same  way  by  the  pro- 
duct of  the  other  two  denominators  (15).  Then  both 
terms  of  \  are  multiplied  by  the  product  of  the  other  two 
denominators  (20). 

Rule  for  reducing  fractions  to  a  common  denomi- 
nator. 

Multiply  both  terms  of  eachjraction  by  the  product  of  all 

the  denominators  except  its  ovm. 
Reduce  i  |  f  to  a  common  denominator. 
Reduce  f  -f-^  and  \\  to  a  common  denominator. 

/Imc    44.0.    8.^4.  and  l-S^.. 

Reduce  ^  1 1  and  ^  to  a  common  denominator. 

Ano   -Li-4.  i^a  S.i5.  and  3^^§- 

-"•"* "288      288      288    """    2  8  8" 

Reduce  f  -^^  and  y\  to  a  common  denominator. 
Reduce  |  f  and  12^  to  a  common  denominator. 

Anv      54     60     888 
.a.11,0.    ij^    ij-^     -^2  • 

Reduce  |  |  and  f  oi  \\  to  a  common  denominator. 

And        7  6_3_     2  5.9  2     XaSil. 
■^""''    345a      3456      3  4  5  6* 


reduction  op  fractions.  167 

Reduction  of  Fhactions  to  theik  Lowest  Terms. 

What  is  the  difference  between  i  and  |  ? 
Ans,  They  express  the  same  value,  by  different  navies. 
Which  fraction  has  the  smallest  numbers  etpployed  to 
express  its  value  ? 

In  the  two  fractions  f  and  /^  is  there  any  difference  in 
the  value  ? 

Which  fraction  has  its  value  expressed  by  the  smallest 
numbers  ? 

A  fraction  is  reduced  to  its  lowest  terms,  when  its  zalue 
is  expressed  by  the  smallest  numbers  which  can  he  used,  to 
express  that  value. 

For  example,  £  is  reduced  to  its  lowest  terms,  because 
no  smaller  numbers  than  3  and  4  can  express  this  value. 
The  value  of  a  fraction  is  not  altered  if  both  terras  of  it 
are  divided  by  the  same  number. 

Thus  if  I  has  both  its  terms  divided  by  2,  it  becomes  f 
and  the  value  remains  the  same.  If  it  is  divided  by  4,  it 
becomes  i  and  its  value  remains  unaltered. 

When  it  was  divided  by  2,  it  was  not  reduced  to  its  low- 

est  terms,  because  smaller  numbers  can  express  the  same 

rvalue  as  |.     But  when  it  was  divided  by  4,  it  was  reduced 

to  its  lowest  terms,   because  no  smaller  numbers  than   1 

and  2  can  express  its  value. 

The  shortest  way  to  reduce  a  fraction  to  its  lowest 
terms  is,  to  divide  it  by  the  largest  number  which  will  di- 
vide both  terms,  without  a  remainder. 

Any  number  which  will  divide  two  or  more  numbers 
without  a  remainder  is  called  a  common  measure,  and  the 
largest  number  which  will  do  this,  is  called  the  greatest 
common  measure. 

In  many  operations  it  saves  much  time  to  have  a  frac- 
tion  reduced  to  its  lowest  terms.  Thus  for  example,  if  we 
are  to  multiply  3429  by  ||  it  would  be  much  easier  to  re- 
duce the  fraction  to  J  (which  are  its  lowest  terms)  and 
then  multiply. 

There  are  many  fractions  which  can  be  reduced  to  their 
lowest  terms  without  much  trouble.  For  example  let  the 
pupil  reduce  these  fractions. 


168  ARITHMETIC.       SECOND  PART. 

Reduce  1 1  J_  jl  _6_  to  their  lowest  terms. 

But  there  are  many  fractions,  which  it  is  much  morv_ 
difficult  to  reduce.  Thus  if  we  wish  to  reduce  yVsV  ***  '^^ 
lowest  terms,  we  could  not  so  readily  do  it. 

In  such  a  case  as  this  there  are  two  ways  of  doing  it  ; 
the  first  is  as  follows. 


Rule  for  reducing  a.  fraction  to  its  lowest  terms. 

Divide  the  terms  of  the  fraction  by  any  number  that  will 
divide  both,  without  a  remainder.  Divide  the  answer  ob- 
tained in  the  same  way.  Continue  thus,  till  no  number  can 
be  found,  that  will  divide  both  terms  without  a  remainder. 

Thus, 

Reduce  /j'_4_.  to  its  lowest  terras. 

N.  B.  The  brackets  at  the  right  of  the  fractions  show 
that  both  terms  of  the  fraction  are  to  be  divided  by  the  di- 
visor,  and  not  the  fraction  itself,  as  in  the  division  effrac- 
tions. 

JL3_4_ \_1_3 78 

1  8  36;^    •   "        6  12 

6  I  2/    "^     3  0  6 

3V6)-^3=-rV2  Ans«>er. 

In  the  above  process,  both  terms  of  the  fraction  y2_3_4_ 
are  divided  by  3  ;  the  answer  is  divided  by  2  ;  and  this 
answer  again  is  divided  by  3. 

The  last  answer  is  jW  which  cannot  have  both  terms 
divided  by  any  number  without  a  remainder. 

The  other  method  of  reducing  a  fraction  to  its  lowest 
terms,  is  first  to  find  the  number  which  is  the  greatest 
common  measure,  and  then  to  divide  the  fraction  by  this 
number. 

The  following  is  the  method  of  finding  the  greatest 
common  measure,  and  reducing  to  the  lowest  terms. 

Reduce  §4  to  its  lowest  terms. 

The  denominator  is  first  placed  as  a  dividend,  and  the 
numerator,  as  a  divisor;  (below.)  After  subtracting,  the 
remainder  (14)  is  used  fnr  the  divisor,  and  the frst  divisor 
(21)  is  used  for  the  dividend.     This  process  of  dividing 


REDUCTION  OP  FRACTIONS.  169 

the  last  divisor  by  the  last  remainder  is  continued  till 
nothing  remains.  The  last  divisor  (7)  is  the  greatest 
common  measure. 

We  then  take  the  fraction  |i  and  divide  both  terms  by 
7,  the  greatest  common  measure,  and  it  is  reduced  to  its 
lowest  terms,  viz.  |. 

21)35(1 
21 


14)21(1 
14 


7)14(2 
14 


00 

Rule  for  finding  the  greatest  common  measure  of 
A  Fraction  and  reducino^  to  its  lowest  terms. 

Divide  the  greater  niimheK,i>y  the  less.  Divide  the  divi- 
sor by  the  remainder,  arid  continue  to  dUfde  the  last  divisor 
by  the  last  remainder,  till  noting  remains.  The  last  divisor 
is  the  greatest  common  measure,  by  which  both  terms  of  the 
fraction  are  to  be  divided,  and  it  is  reduced  to  its  lowest 
terms. 

Reduce  the  following  Fractions  to  their  lowest  tenns. 

486   .   144   .  324.   14  29.  16  4  4..  _4_6_8_.  _47_4_6_  •  A^^-f    '. 
'OTa'O  '  T728'  '  6  4  8'  ¥8  58'  2192'  1184'  38433'  42315  ' 


Rprliipp  thp  fnllnwino-  •    386.    4  9  33.    _12_3JL5_  •    JU_6_8  7 

iveauce  me  louowiiig  .  -^-^j  ,    87-84  j   eTagio  '    343954 
'■>  1 
15 


94  8 
_9.9.8.8il       •     _1_0_32_8_4_ 

9998881T'     7  3  28473 


170  ARITHMETIC.       SECOND    PART. 

REDUCTION  OF  FRACTIONS  FROM  ONE  ORDER 
TO  ANOTHER  ORDER. 

It  will  be  recollected  that  in  changing  whole  numbers 
from  one  order  to  another,  it  was  done  by  multiplication 
and  division. 

Thus,  if  40  shillings  were  to  be  changed  to  pounds,  we 
divided  them  by  the  number  of  shilUngs  in  a  pound,  and  if 
£2  were  to  be  reduced  to  shillings,  we  midtiplied  them  by 
the  number  of  shillings  in  a  pound. 

The  same  process  is  used  in  changing  fractions  of  one 
order  to  fractions  of  another  order. 

Thus,  if  we  wish  to  change  ^i^  of  a  £  to  a  fraction  of 
the  shilling  order,  we  multiply  it  by  20,  making  it  g^"^. 
For  2Y0  °^^  shilling  is  the  same  as  ^^o  of  a  pound. 

If  we  wish  to  change  -^W  of  a  shilling,  to  the  same 
value  in  a  fraction  of  the  pound  order,  we  divide  ^Vo  "^y 
20,  making  it  ■^^^.  (This  could  also  be  divided  by  multi- 
plying its  denominator  by  20.) 

If  then  we  wish  to  change  a  fraction  of  a  lower  order  to 
the  same  value  in  a  higher  order,  we  must  divide  the  frac- 
tion,  by  multiplying  the  denominator,  by  that  number  of  units 
(of  the  order  to  which  the  fraction  belongs)  which  make 
a  unit  of  the  order  to  which  it  is  to  be  changed. 

Thus  if  we  wish  to  change  |  of  a  penny  to  the  same 
value  in  the  fraction  of  a  shilling,  we  multiply  its  denomi- 
nator by  12,  making  it  ^\  of  a  shilling.  If  we  wish  to 
change  this  to  the  same  value  in  a  fraction  of  the  pound 
order,  we  must  now  multiply  its  denominator  by  the  num- 
ber of  shillings  which  make  a  pound,  making  it  y/2  0  ^^  ^ 
pound.  It  must  be  remembered  that  multiplying  the  deno- 
minator of  a  fraction,  is  dividing  the  fraction. 

If,  on  the  contrary,  we  wish  to  change  a  fraction  of  a 
higher  order  to  one  of  the  same  value  in  a  loiver  order,  we 
must  multiply. 

Thus,  to  change  y|^  of  a  shilling  to  the  penny  order, 
we  must  multiply  it  by  12.  This  we  do,  by  multiplying 
its  numerator  by  12,  and  the  answer  is  y^T •  ^^'^  ''^^  there 
are  12  times  as  many  whole  pence  in  a  whole  shiUing,  so 
there  are  12  times  as  many  yl^-  of  a  penny  in  y|y  of  a 
shilling. 


reduction  of  fractions.  171 

Rule  for  reducing  fractions  of  one  order  to  an- 
other ORDER. 

To  reduce  a  fraction  of  a  higher  to  one  of  a  lower 
order. 

Multiply  the  fraction  by  that  number  of  units  of  the  next 
lower  order,  tvhich  are  required  to  make  one  unit  of  the  order 
to  which  the  fraction  belongs.  Continue  this  process  till  the 
fraction  is  reduced  to  the  order  required. 

To  reduce  a  fraction  of  a  lower  to  one  of  a  higher 
order. 

Divide  the  fraction  (by  multiplying  the  denominator)  by 
the  number  of  imiis  which  are  required  to  make  one  unit  of 
the  next  higher  order.  Continue  this  process  till  the  frac- 
tion is  reduced  to  the  order  required. 

Examples. 

Reduce  yaV?  of  a  guinea,  (or  of  28  shillings,)  to  the 
fraction  of  a  penny. 

Reduce  a  of  a  guinea  to  the  fraction  of  a  pound. 

Reduce  -^\  of  a  pound  Troy,  to  the  fraction  of  an 
ounce. 

Reduce  -^^  of  an  ounce  to  the  fraction  of  a  pound  Troy. 

Reduce  ^^  of  a  pound  avoirdupoise  to  the  fraction  of  an 
ounce. 

A  man  has  jr^j  of  a  hogshead  of  wine,  what  part  of  a 
pint  is  it  ? 

A  vine  grew  ^fj-^  of  a  mile,  what  part  of  a  foot  was  it  ? 

Reduce  |  of  |  of  a  pound  to  the  fraction  of  a  shilling. 

Reduce  f  of  |  of  3  shillings,  to  the  fraction  of  a  pound. 


REUCTION  OF   FRACTIONS   OF   ONE  ORDER, 
TO  UNITS  OF  A  LOWER  ORDER. 

It  is  often  necessary  to  change  a  fraction  of  one  order, 
to  units  of  a  lower  order.  For  example,  we  may  wish  to 
change  |  of  a  unit  of  the  pound  order,  to  units  of  the 
shilling  order. 

This  I  of  a  £  is  2  pounds  divided  by  3.     These  2  pounds 


172  ARITHMETIC.   SECOND  PART 

are  changed  to  shillings,  by  multiplying  by  20,  and  then 
divided  by  3,  and  the  answer  is  13^  shillings.  This  i  of 
a  shilling  maybe  reduced  to  pence  in  the  same  way,  for  \ 
of  a  shilling  is  1  shilling  divided  by  3.  This  1  shilling  can 
be  changed  to  fence,  and  then  divided  by  3,  the  answer  is 
4  pence. 


Rule  for  finding  the  Value  of  a  Fraction  in  units 
OF  a  lower  order. 

Consider  the  numerator  as  so  many  units  of  the  order  in 
which  it  stands,  and  then  change  it  to  units  of  the  order  in 
which  you  wish  to  find  the  value  of  the  fraction.  Divide 
by  the  denominator,  and  the  quotient  is  the  answer,  and  is  of 
the  same  order  as  the  dividend. 

Examples. 

1.  How  many  ounces  in  |  of  a  lb.  Avoirdupoise  ? 

2.  How  many  days,  hours  and  minutes,  in  4  of  a  month  1 

3.  What  is  the  value  of  f  of  a  yard  ? 

4.  What  is  the  value  of  /_  of  a  ton  ? 

5.  How  many  pence  in  |  of  a  lb.  ? 

6.  How  many  drams  in  f  of  a  lb.  avoirdupoise  ? 

7.  How  many  grains  in  f  of  a  lb.  Troy  weight  ? 

8.  How  many  scruples  in  |  of  a  lb.  Apothecaries 
weight  ? 

9.  How  many  pints  in  f  of  a  bushel  I 


REDUCTION  OF  UNITS  OF  ONE  ORDER  TO 
FRACTIONS  OF  ANOTHER  ORDER. 

It  is  necessary  often  to  reverse  the  preceding  process, 
and  change  units,  to  fractions  of  another  order.  For  ex- 
ample, to  change  1 3s.  4d.  to  a  fraction  of  the  pound  or- 
der. 

To  do  this  we  change  the  13s.  4d.  to  units  of  the  lowest 


REDUCTION  OF  FRACTIONS. 


173 


order  mentioned,  viz.  160  fence.  This  is  to  be  the  numer- 
ator of  the  fraction.  We  then  change  a  unit  of  the  pound 
order  to  pence  (240)  and  this  is  the  denominator  of  the 
fraction.     The  answer  is  i|^  of  a  pound. 

For  if  13s.   4d.  is  160  pence,  and  a  lb.  is  240  pence, 
then  13s.  4d.  is  |-|^  of  a  pound. 


RULK  FOR  REDUCING  UNITS  OF  ONE  ORDER  TO  FRACTIONS 
OF  ANOTHER  ORDER. 

Change  the  given  sum  to  units  of  the  lowest  order  men- 
tioned, and' make  them  the  numerator. 

Change  a  unit  of  the  order  to  which  the  sum  is  to  he  re- 
duced, to  units  of  the  same  order  as  the  numerator,  and  place 
it  for  the  denominator. 

Examples. 

Reduce  6oz.  4pwt.  to  the  fraction  of  a  pound  Troy. 
Reduce  3  days,  6  hours,  9  minutes  to  the  fraction  of  a 
month. 

Reduce  2cwt.  2qrs.  161bs.  to  the  fraction  of  a  ton. 
Reduce  21b.  4oz.  to  the  fraction  of  a  cwt. 


REDUCTION  OF  A  COMPOUND  NUMBER  TO  A 
DECIMAL  FRACTION.  , 

It  is  often  convenient  to  change  a  compound  number,  to  a 
decimal  fraction. 

Thus  we  can  reduce  loz.  lOpwt.  to  a  decimal  of  tJie 
pound  order. 

Let  the  figures  be  placed  thus,  and  the  process  will  be 
explained  below.  The  10  pwts.  are  first  written,  and  then 
the  1  oz.  set  under. 

20)10'0  pwt. 
12)  1'5  oz. 
a25lb. 
15* 


174  ARITHMETIC.      SECOND  PART. 

We  first  change  the  lowest  order  (10  pwts.)  to  an  ira- 
proper  decimal,  thus  lO'O.  Now  as  20  pv/ts.  make  an  oz., 
there  are  but  one  twentieth  as  many  ounces  in  a  sum  as 
there  are  penny  weights. 

For  the  same  reason,  in  any  sum  there  are  but  one  twen- 
tieth  as  many  tenths  of  an  ounce  as  there  are  tenths  of  a 
penny  weight. 

As  there  are  then  100  tenths  ofapwt.  in  this  sum,  if  we 
take  one  twentieth  of  them,  we  shall  find  how  many  tenths 
of  an  oz.  there  are. 

We  therefore  divide  the  lO'O  pwts.  by  20,  and  the 
amount  is  ,5.  This  ,5  is  placed  (beside  the  1  oz.  of  the 
sum)  under  the  lO'O  pwts.,  and  thus,  instead  of  reading 
the  sum  as  loz.  10  pwts.,  we  read  it  as  1,5  oz.,  or  loz. 
and  5  tenths  of  an  oz. 

As  the  pwts.  are  thus  reduced  to  the  decimals  of  an  oz. 
we  now  reduce  the  l,5oz.  to  the  decimal  of  a  lb.  in  the 
same  way. 

We  make  the  1,5  an  improper  decimal,  thus  1'5  (15 
tenths)  of  an  oz. 

Now  as  there  are  12oz.  in  a  lb.,  there  are  but  one  twelfth 
as  many  tenths  of  a  lb.  in  a  sum,  as  there  are  tenths  oj  an 
oz.  We  therefore  divide  the  15  tenths  of  an  oz.  by  12, 
and  the  answer  is  ,1  of  a  lb.  and  3  left  over.  This  3  is  re- 
duced  to  hundredths  by  adding  a  cipher  and  dividing  it 
again.  The  quotient  is  2  hundredths.  The  next  remain- 
der is  changed  to  thousandths  in  the  same  way,  and  the 
answer  is  ,125  of  a  £. 


Rule   for  changing  a  compound  kumbek  to  a  deci- 
mal. 

Change  the  loioest  order  to  an  improper  decimal.  Divide 
it  by  the  number  of  units  of  this  order,  which  are  required 
to  make  a  unit  of  the  next  higher  order,  and  set  the  answer  be- 
side the  units  of  the  next  higher  order.  Repeat  this  process 
till  the  sum  is  brought  to  the  order  required. 

Examples. 
Reduce  10s.  4d.  to  the  decimal  of  a  lb. 


REDUCTION  OF  FRACTIONS.         175 

Reduce  8s.  Gd.  3qrs.  to  the  decimal  of  a  lb. 
Reduce  17hrs.  16min.  to  the  decimal  of  a  day. 
Reduce  3qrs.  2na.  to  the  decimal  of  a  yd. 
Reduce  32gals,  4qts.  to  the  decimal  of  a  hogshead. 
Reduce  lOd.  3qrs.  to  the  decimal  of  a  shilling. 


t      REDUCTION  OF  A  DECIMAL,  TO  UNITS  OF 
COMPOUND  ORDERS. 

The  preceding  process  can  be  reversed,  and  a  decimal 
of  one  order,  be  changed  back  to  units  of  other  orders. 

Thus,  if  we  have  ,125  of  a  lb.  Troy,  we  can  change  it 
to  units  of  the  oz.  and  pwt.  order. 

In  performing  the  process,  we  place  the  figures  thus. 

,1251b. 
12 


l,500oz. 
20 

10,000pwt. 

We  reason  thus.  In  ,125  of  a  lb.  there  must  be  12 
times  as  many  thousandths  of  an  oz.  (for  12  oz.  =  1  lb. 
We  therefore  multiply  by  12,  and  point  off  according  to 
rule,  and  the  answer  is  1  oz.  and  500  thousandths  of 
an  oz. 

Now  as  we  have  found  how  many  oz.  there  are,  we 
must  find  how  many  pwts,  there  are  in  the  ,500  of  an  oz. 
There  must  be  20  times  as  many  thousandths  of  a  pwt.  as 
there  are  thousandths  of  an  oz.  therefore  multiply  the  de- 
cimal only,  by  20,  and  point  off  according  to  rule,  and  we 
find  there  are  10  pwts. 

We  have  thus  found  that  in  ,125  of  a  lb.  there  are  loz. 
and  lOpwts. 


176  arithmetic.     second  part. 

Rule  for  changing  a  decimal  of  one  compound  or- 
der, TO  UNITS  of  other  ORDERS. 

Multiply  the  decimal  by  the  number  of  units  of  the  next 
lower  order  which  are  required  to  make  one  unit  of  the  order 
in  which  the  decimal  stands. 

Point  off  according  to  rule,  and  multiply  the  decimal  part 
of  the  anstver  in  the  same  way,  pointing  off  as  before.  Thus 
till  the  sum  is  brought  into  the  order  required.  The  units 
of  each  answer  make  the  final  answer. 

In  ,1257  of  a  ^  how  many  shillings,  pence  and  farth-  ^ 
ings  ? 

What  is  the  value  of  ,2325  of  a  ton  ? 
What  is  the  value  of  ,375  of  a  yard  ? 
What  is  the  value  of  ,713  of  a  day  ? 
What  is  the  value  of  ,15834821  of  a  ton  ? 


REDUCTION  OF  CURRENCIES. 

There  are  few  exercises  in  Reduction,  of  more  prac 
tical  use  than  the  Reduction  of  Currencies,  by  which  a 
sum  in  one  currency  is  changed  to  express  the  same  val- 
ue in  another  currency. 

An  example  of  this  kind  of  reduction  occurs,  when  the 
value  of  il  is  expressed  in  British  currency  thus,  43.  6d. 

The  necessity  for  using  this  process  in  this  country,  re- 
sults from  the  following  facts. 

Before  the  independence  of  the  U.  States,  business  was 
transacted  in  the  currency  of  Great  Britain.  But  at  vari- 
ous times,  the  governments  of  the  different  States,  put 
bills  into  circulation,  which  constantly  lessened  in  value, 
until  they  became  very  much  depreciated.  For  example, 
a  bill  which  was  called  a  pound  or  twenty  shillings,  British 
currency,  was  reduced  to  be  worth  only  fifteen  shillings, 
in  the  New  England  states. 

This  depreciation  was  greater  in  some  states  than  it 
was  in  others,  and  the  result  is,  that  pounds,  shillings,  and 
pence  have  different  values  in  different  states. 

12  pence  make  a  shilling,  and  20  shillings  make  a 
pound,  in  all  cases,  but  the  value  of  a  penny,  a  shilling, 
or  a  pound,  depends  upon  the  currency  to  which  it  be- 
longs. 


REDUCTION  OF  CURRENCIES. 


177 


The  following  table  shows  the  relative  value  of  the  sev- 
eral  currencies,  by  showing  the  value  of  one  dollar  in  each 
of  the  different  currencies. 

VALUE  OF  ONE  DOLLAR  IN  EACH  OF  THE  DIFFERENT  CUU- 
RENCIES. 

equals  6.?.  New  England  currenc}'. 
"       8s.  New  York  currency. 
"       7s.  6d.  Pennsylvania  currency. 
"       4*.  8c/.  Georgia  currency. 
"       4.S.  6d.  Sterling  money,  or  Eng.  currency. 
"       5s.  Canada  currency. 
"       4:S.  10\d.  Irish  currency. 

"  £2. 14s.  Scotch  currency. 

VALUE  OF  ONE  POUND  OF  EACH  OF  THE  DIFFERENT  CUR- 
RENCIES, EXPRESSED  IN  FEDERAL  MONEY. 

£1  N.  England  currency  equals     83,3331 
£1  N.  York  currency  "         -1^2,50 

£1  Pennsylvania  currency    "         §2,G66| 
£1  Georgia  currency  "         M,285^ 

£1  Sterling  money  "         $4,444^ 

£1  Canada  currency  "         84,00 

£1  Irish  currency  "         84,102- 

£1  Scotch  currency  "         80,3701  ii 

The  following  sums  for  mental  exercise,  will  be  found 

of  much  practical  use,  and  should  be  practised  till  tliey  can 

be  readily  answered. 

Examples  in  N.  England   currency  for  mental  ex- 
ercise. 

1.  If  6  shillings  equal  a  dollar  or  100  cts.  how  many 
cents  in  3  shillings  ?  in  2  shillings  ?  in  1  shilling  ?  in  4 
shillings  ?  in  5  shillings  ? 

2.  If  1  shilling  is  16|  cts.  how  many  cents  in  6  pence? 
in  3  pence  ?  in  9  pence  ?  in  4  pence  ?  in  7  pence?  in 
8  pence  ?  in  11  pence  ? 

3.  How  many  cents  in  Is.  6d.  ?  in  Is.  9d.  ?  in  Is.  3d.? 
in  2s.  6d  ?  in  2s.  9d.  ?  in  3s.  4d.  ?  in  5s.  6d.  ?  in  7s.  6d.? 


178  ARITHMETIC.       SECOND  PART. 

in  8s.  6d.  ?  in  9s.  ?  in  9s.  Gd.  ?    in  10s.  6d.  ?  in  lis.  ?  in 
lis.  6d.  ?  in  12s.  ? 

4.  If  6d.  is  81  cts.  how  many  cents  is  3d.  ?  how  many- 
is  Id.  ?  now  many  is  2d.  ? 

5.  If  you  buy  8  yds.  of  ribbon  at  Is.  Od.  yer  yd.  how 
much  will  the  whole  cost  ? 

6.  If  you  buy2|  yds.  of^muslin  at  2g.  6d.  per  yd.  how 
much  will  it  cost  in  dollars  and  cents  ? 

7.  If  you  buy  3i  yds.  of  ribbon  at  Is.  9d.  per  yd.  how 
much  will  it  cost  ? 

8.  If  you  buy  a  brush  for  2s.  3d.  and  a  penknife  for 
4s.  6d.  and  a  comb  for  Is.  6d.  how  much  is  given  for  the 

whole  ? 

9.  If  you  pay  3s.  6d.  for  scissors,  2s.  4d.  for  a  thimble, 
and  Is.  9d.  for  needles,  how  much  will  the  whole  cost  ? 

10.  If  linen  is  4s.  6d.  per  yd.  how  much  will  4f  yds. 
cost  ? 

11.  If  apiece  of  calico  is  2s.  3d.  per  yd.  how  much  will 
6i  yds.  cost  ? 

12.  If  muslin  is  4s.  6d.  per  yard,  what  will  2J  yds  cost  ? 

13.  How  much  is  lUd.?  lOid.  ?  9id.  ?  Sid.  ?  7id.  1 
12^d. ?  16id. ? 

Examples  in  N.  York  ovrrency  for  mental  exercise. 

1.  If  a  dollar  in  N.  York  currency  is  8s.  how  many 
cents  in  4s.  ?  in  2s.  ?  in  Is.  ?  in  5s. '(  in  6s.  ?  in  7s.  ?  in 
9s.  ?  in  10s.  1  in  lis.  ?  in  12s.  ?  in  13s.  ?  in  14s.  ?  in 
15s.?  in  16s,? 

2.  If  onfe  shilling  is  12^  cts,  how  many  cents  in  6d.  ?  in 
3d.  ?  in  Id.  ?  in  2d.  1  in  4d.  ?  in  7d.  ?  in  8d,  ?  in  9d.  ? 
in  lOd.  ?  in  lid.? 

3.  How  many  cents  is  Is.  6d.  N.  York  currency  ?  is 
2s.  6d,  1  is  3s.  6d.  ?  is  5s.  3d.  ?  is  6s.  9d.  ?  is  4s.  8d.  ? 

Questions  can  be  asked  in  the  other  currencies  in  the 
same  manner. 


REDUCTION    OF   CURRENCIES.  179 

REDUCTION  OF  CURRENCIES  TO  FEDERAL 
MONEY. 

Sums  of  this  kind,  which  are  too  complicated  to  be 
done  mentally,  may  be  performed  on  the  slate,  by  the 
following  rules. 

To  REDUCE  BRITISH  GURRErJCY  TO  FeDERAL  MoNEY. 

Reduce  the  sum  to  a  decimal  of  the  pound  order,  and  di- 
vide the  ansioer  by  ■^\. 

The. reason  of  tliis  rule  is  that  a  dollar  is  /-  of  a  £  of 
this  currency,  and  therefore  there  are  as  many  dollars  in 
the  sum  as  there  are  /„  in  it. 

Note.  Before  reducing  any  currency  to  Federal  mon 
ey,  the  sum  must  he  reduced  to  a  decimal  of  the  pound  or' 
der.     After  this  process  the  following  rules  may  be  used. 

To  REDUCE  Canada  currency. 

As  a  dollar  is  i  of  a  £  in  this  currency,  there  will  be  as 
many  dollars  as  there  are  {  in  the  sum.     Therefore, 
Reduce  the  sum  to  the  decimal  of  a  £  and  divide  it  by  I. 

To  reduce  New  England  Currency. 

As  1  dollar  is  ,3  of  a  pound  in  this  currency,  so  there 
are  as  many  dollars  in  a  sum  of  N.  England  currency  as 
there  are  ,3  in  it.     Therefore 

Reduce  the  sum  to  the  decimal  of  a  £  and  divide  it  by  ,3. 

To  reduce  New  York  Currency. 

As  1  dollar  is  ,4  of  a  pound  in  this  currency,  there  will 
be  as  many  dollars  in  a  sum  of  New  York  currency,  as 
there  are  ,4  in  it.     Therefore 

Reduce  the  sum  to  the  decimal  of  a  £  and  divide  it  by  ,4. 

To  reduce  Pennsylvania  Currency. 

As  1  dollar  is  |  of  a  £  in  this  currency  there  are  as 
many  dollars  in  the  sum  as  there  f  contained  m  it. 
Therefore 

Reduce  the  svm  to  the  decimal  of  a  £  and  divide  U  by  f . 


180  ARITHMETIC.       SECOND  PART. 

To  REDUCE  Georgia  Currency. 

As  1  dollar  is  ^^  of  a  pound  in  this  currency  there  are 
as  many  dollars  in  the  sum  as  there  3'^  contained  in  it. 
Therefore 

Reduce  to  the  decimal  of  a  £  and  divide  the  sum  by  /^ . 


REDUCTION  OF  FEDERAL  MONEY  TO  THE 
SEVERAL  CURRENCIES. 

To  change  a  sum  in  federal  money  to  the  different  cur- 
rencies, the  preceding  process  is  reversed,  and  the  sum  is 
to  be  multiplied  (instead  of  divided)  by  the  several  frac- 
tions.  The  answer  is  found  in  pounds  and  decimals  of  a 
pound.  The  decimal  can  be  reduced  to  units  of  the  shil- 
ling and  pence  order  by  a  previous  rule.  (p.  176.) 

Examples. 

1.  Reduce  Is.  6d.  in  the  several  currencies  to  Federal 
money. 

Answers. 

Of  Canada  Currency,  it  is  $,30 

British,  "  $,333i 

N.  England,  "  $,25 

N.  York,       «  $,187i 

Penn.  «  $,20 

Georgia,        "  $,321f 

2.  Reduce  4id.  of  the  several  currencies  to  Federal 
money. 

3.  Reduce  4s.  6d.  of  the  several  currencies  to  Federal 
money. 

4.  Reduce  35£  3s.  7id  of  the  several  currencies  to 
Federal  money. 

5.  Reduce  $118,25  to  the  several  currencies. 


H 


KEDrCTION  OF  CUBBENCIES.  181 

Answer?  to  the  last. 


In  Canada  currency,  it  is 
British,  " 

N.  Eng.  " 

N.  York, 
Penn.  " 

Georgia,  " 

Reduce  2s.  9d.  of  N.  England  currency  to  the  same 
value  in  all  other  currencies. 

Reduce  4s.  6d.  N.  York  currency  to  the  same  value  in 
all  the  other  currencies. 


£ 

s.   d. 

29  " 

11  "  3 

26  " 

12  "  li 

^5  " 

9  «  6 

47  " 

6  "  0 

44  " 

6  "  lOi 

27  " 

11  "  9f 

REDUCTION  FROM  ONE  CURRENCY  TO  AN- 
OTHER. 

The  following  table  will  enable  the  pupil  to  reduce  a 
sum  from  one  currency  to  another,  with  more  facility  than 
by  any  other  method.  Each  fractional  figure  shows  the 
relative  value  of  a  sum  in  one  currency  to  the  same  sum 
in  another  currency. 

For  example,  the  |  in  the  second  perpendicular  and  the 
fourth  horizontal  column,  shows  that  £1  sterling  is  |  of  the 
number  which  expresses  the  same  value  in  New  England 
currency.  Thus  £6  sterling  is  f  of  the  number  which 
expresses  the  same  value  in  New  England  currency.  That 
is,  £6  is  J  of  the  answer  to  be  obtained  when  the  same 
value  is  expressed  in  New  England  currency.  To  find 
the  answer^  we  reason  thus.  If  6£  is  three  fourths,  £2  is 
one  fourth,  and  8£  is  the  answer.     Thus  dividing  by  2. 

Rule  for  changing  a  sum  in  one  currency,  to  the 

SAME  value  in  ANOTHER  CURRENCY. 

To  change  a  sum  in  a  currency  written  in  the  upper  space 
to  one  written  in  the  right  hand  space,  divide  by  the  fraction 
that  sta7ids  where  both  spaces  meet. 

If  there  are  shillings,  pence  and  farthings  in  the  s«m, 
first  reduce  them  to  the  decimal  of  a  £. 

16  , 


182 


ARITHMETIC.       SECOND  PART. 


TABLE 

EXHIBITING  THE  COMPARATIVE  VALUES  OF  THE  SEVERAL  CURRENCIES. 


ANY  SUM  EXPRESSED  IN 

1 

it 

b5 

o 

CD 
O 

b5 
o 

3 

b5 

B 

b5 

CO 

o 
o 

IS 

o 
S  ! 

% 

K 

e 

B 

■c 

B 

E 

o 

1  0 
2T 

1 

1  2 

7 
8  1 

_2_S_0_ 

2  76  9 

_5_ 
54 

1 

9 

3^ 

A> 

£  Scot. 

I   0 
4 

A 

7 
1  2 

±125 

1846 

5. 
8 

3 
4 

16 
1  5 

27 
4 

£N.Y. 

8 
3 

3 

5 

2  8 
45 

600 
923 

2 
3 

4 
5 

16 
I  5 

3,6 
5 

£  Pen. 

y 

3 

4 

1 

75  0 
92  3 

i 

5 
4 

4 
3 

9  limes 

£N.E. 

4  times 

._9_ 
1  0 

14 
1  S 

9  00 
923 

6 
5 

3 
2 

8 
5 

5  4 
5 

£Can. 

Iff 

93J?_ 
10  0  0 

6461 
6  7  5  0 

9.23 
9  0  0 

92  3 

750 

92  3 

COO 

1846 
112  5 

2  7  69 
25  0 

£  Irish. 

3_0 

H 

67  50 
6  4  6  1 

J.5 
1  4 

9 

7 

4.5. 
2  8 

\J 

8  1 

7 

£Geo. 

V 

i  If 

10  00 
923 

V 

4 
3 

s. 

3 

V     1  times 

1 

£  Ster. 

/o 

7 
3  0 

225. 
92  3 

1 

4 

.JL 
1  0 

3 

8 

4_     1     27 

To    '     I  0 

$  F.M. 

EXAMPLES  FOR  PRACTICE. 


1.  Reduce  £4  N.  E.  to  F.  M. 

2.  Reduce  2£  3s.  9d.  N.  E.  to  F.  M. 

3.  Reduce  £6  N.  Y.  to  F.  M. 

4.  Reduce  £8  ;  4  ;   9  N.  Y.  to  F.  M. 

5.  Reduce  £3  ;  2  ;  3  Penn.  to  F-  M 

6.  Reduce  $152.60  to  N.  E. 

7.  RedCice  $196.00  to  N.  E. 
rt.  Reduce  $629.00  to  N.  Y. 


Ans.  f  1.S..S331. 

Ans.  $7,291 1. 

Ans.  $15.00. 

Ans.  $20.593f . 

Ans.  $8.30. 

Ans.  £45  ;   15  ;  7.2. 

Ans.  $58  ;   16. 

Ans.  251  ;   12. 


g!  Reduce  £35  ;  6  ;  8  sterling  to  N.  E 

A71S.  £47  ;  2  ;  2  ;  2| 

10.  Reduce  £120  N.  E.  to  Can.  Ans.  £100 

11.  Reduce  £155  ;  13  N.  E.  to  Sterling. 

Ans.  £116;  14;  9 


BEDUCTIOK   OP   cnRRENCIE3.  183 

12.  Reduce  £104  ;  10  Can.  to  N.  Y.    Am.  £167  ;  4. 

13.  Reduce  £300  ;  10  ;  4  ;  2  Can.  to  Penn. 

Ans.  £450  ;  15  ;  6  ;  3. 

14.  Reduce  £937  ;  18;  11  ;  1  N.  E.  to  Geo. 

Ans.  £721 ;   14  ;  8  ;  3. 

15.  Reduce  $224  ;  60  to  Can.  Ans.  £56  ;  3. 

16.  Reduce  £225  ;  6  N.  E.  to  F.  M.  Ans.  $752.00. 

17.  Reduce  £880  15  ;   11  ;   1  Penn.  to  Sterling. 

Ans.  528  9  ;  6  ;   3. 

18.  Reduce  £6,750  Irish  to  Geov.  Ans.  £6,461. 

19.  Reduce  £1,846  Ster.  to  Irish.  Arts.  £2,000. 

20.  Reduce  £1,722  ;   18  ;   9  ;  3  N.  E.  to  N.  Y. 

Ans.  £2,298  ;    5  ;   1. 

21.  Reduce  £2,114  ;  1  ;  3  Can.  to  F.  M. 

Ans.  $8,456.25. 

22.  Change  £784  ;  5  ;  6  ;  2  Penn.  to  Geor. 

Ans.     £487;   19;   10  rHh 

23.  Change  £923  SterUng  to  Irish. 

24.  Change  £,4000  Irish  to  SterUng. 

25.  Change  £157  ;  8  ;  3  ;  3  N.  Y.  to  N.  E. 

26.  Change  £1,654  ;  3  ;  8  ;   1  Penn.  to  N.  E. 

27.  Change  £  947  ;  9  ;  4  ;  2  N.  E.  to  F.  M. 

28.  Change  $1,444.66  to  N.   E.    To  N.  Y.      To  Penn. 

29.  Change  $945.22  to  N.  Y.     To  Geor.     To  Can. 

30.  Change  £1,846  ;    15  ;  4  N.  E.  to  F.  M.      To  Penn 

To  Georgia. 

31.  Change  $4,444,444^  to  Sterling. 

32.  Reduce  £1,000,000  Sterling  to  F.  M. 


ARITHMETIC. 

THIRD  PART. 


NUMERATIOiV. 

In  the  following,  Third  Part,  there  will  be  a  reveiw  of 
the  preceding  subjects,  embracing  the  more  difficult  ope- 
rations. The  rules  and  explanations  will  not  be  repeated, 
as  the  pupils  can  refer  to  them  in  the  former  part. 


ROMAN    NUMERATION. 

Before  the  introduction  of  the  Arabic  figures,  a  method 
of  expressing  numbers  by  Roman  Letters  was  employed. 
As  this  method  has  not  entirely  gone  out  of  use,  it  is  im- 
portant  that  it  should  be  learned.  The  following  letters 
are  employed  to  express  numbers. 

I.  One.  X.  Ten. 

II.  Two.  L.  Fifty. 

III.  Three.  C.  One  Hundred. 

nil.  or  IV.  Four.  D.  Five  Hundred. 

V.  Five.  M.  One  Thousand. 

The  above  letters,  by  various  comhinnlions,  are  made 
to  express  all  the  numbers  ever  employed  in  Roman  Nu- 
meration. 


RULE    FOR    WRITING    AND    READING    ROMAN    NUMBERS. 

As  often   as   a   letter  is  repeated,  its  value  is  repeated. 
When  a  less  number  is  put  before  a  greater,  the  less  number 
is  subtracted.  But  when  the  less  number  is  put  after  the  great'  ■ 
er,  it  is  added  to  the  greater. 

.  Examples.  In  IV.  the  less  number  I.  is  put  before  the 
greater  number  V.  and  is  to  be  subtracted,  making  the 
number  four. 

In  VI.  the  less  number  is  put  after  the  greater,  and  it  is 
to  be  added,  making  the  number  six. 


NUMERATION, 


185 


In  XL.  the  ten  is  subtracted  from  the  fifty. 

In  LX.  the  ten  is  added  to  the  fifty. 

The  following  is  a  table  of  Roman  Numeration  : 

TABLE. 
One 
Two 
Three 
Four 
Five 
Six 
Seven 
Eight 
Nine 
Ten 
Twenty 
Tliirty 
Forty 
Fifty 
Sixty 
Seventy 
Eighty 

*  Iq.  is  used  instead  of  D.  to  represent  five  hundred,  and  for  every  additional 
3.  annexed  at  the  right  hand,  the  number  is  increased  ten  times. 

t  Clg.  is  used  to  represent  one  thousand,   and  for  every  C.  and  g.  put  at  each 
end,  the  number  is  increased  ten  times. 

X  A  line  over  any  number  increases  its  value  one  thousand  timts. 


I. 

Ninety 

LXXXX.  orXC 

II. 

One  hundred 

C. 

III. 

Two  hundred 

CC. 

nil.  or  IV. 

Three  hundred 

CCC. 

V. 

Four  hundred 

CCCC. 

VI. 

Five  hundred 

D.  or  lo* 

VII. 

Six  hundred 

DC. 

VIII. 

Seven  hundred 

DCC. 

vim.  or  IX. 

Eight  hundred 

DCCC. 

X. 

Nine  hundred 

UCCCC. 

XX. 

One  thousand 

M.  or  CiD.t 

XXX. 

Five  thousand 

loo- or  v.t 

XXXX.orXL. 

Ten  thousand 

<-'<^I03-  or  X. 

L. 

Fifty  thousand 

laoa- 

LX. 

Hundred  thousand CCCl330.  or  C 

LXX. 

One  million 

IVl 

LXXX. 

Two  million 

ftlM 

Write  the  following  numbers  in  Roman  letters  : 

5.  7.  3.  9.  8.  IG.  4.  14,  5,  15.  G.  16. 
'26.  36.  30G.  1.  11.  ill,  7,  17.  77.  777. 
1800.     1832.     1789. 

Read  the  following  Roman  numbers  : 

VI.  XIX.  XXIV.  XXXVI.  XXIX.  LV.  XLI. 
LXIV.     LXXXVIII.    XCIX.     MDCCCXVIII, 


OF  OTHER  METHODS  OF  NUMERATION. 

By  the  common  method  of  numeration,  ten  units  of  one 
order,  make  one  unit  of  the  next  higher  order.  But  it  is 
equally  practicable,  to  have  any  other  number  than  ten, 
to  constitute  a  unit  of  a  higher  order.  Thus  we  might 
have  six  units  of  one  order  make  one  unit  of  the  next 
higher  order.  Or  twelve  units  of  one  order  might  make 
one  of  the  next  higher  order. 

The  number  which  is  selected  to  constitute  units  of  the 
higher  orders,  is  called  the  radix  of  that  system  of  nu^ 
meration.  16. 


186  ARITHMETIC.       THIRD  PART. 

The  radix  of  the  common  system  is  ten,  and  this  num- 
ber it  is  supposed  was  selected,  because  men  have  ten 
fingers  on  their  hands,  and  probably  used  them  in  express- 
ing numbers. 

Before  the  introduction  of  the  Arabic  figures,  Ptolemy 
introduced  a  method  of  numeration,  in  which  sixty  was 
the  radix.  The  Chinese  and  East  Indians  use  it  to  this 
day. 

But  in  Ptolemy's  system  there  were  not  sixty  different 
characters  employed.  Instead  of  this,  the  Roman  method 
of  numeration  was  used  for  all  numbers  as  far  as  sixty, 
and  then  for  the  next  higher  orders  the  same  letters  were 
used  over  again,  with  an  accent  (')  placed  at  the  right. 
For  the  third  order  two  accents  (")  were  used,  and  for  the 
fourth  order  three  accents  ('"). 

To  illustrate  this  method  by  Arabic  figures,  31'  23 
signifies  31  sixties  and  23. 

We  have  some  remnants  of  this  method  in  the  division 
of  time  into  60  seconds  for  a  minute,  and  60  minutes  for 
an  hour,  and  also  the  division  of  the  degrees  of  a  circle, 
into  60  seconds  to  a  minute,  and  60  minutes  to  a  degree. 


EXI^RCISES    IN    NUMERATION,    COMMON,    VULGAR,    AND 
DECIMAL. 

{See  rules  on  pages  53,  58,  and  64.) 

1.  Two   million,    four  thousand,   one   hundred   and 
six. 

2.  Two  hundred  thousand,  and  six  tenths. 

3  Twenty  six  billion,  six   thousand,   and  fifteen  thou, 
sandths. 

4.  Two  hundred  and  sixty  thousand  millionths. 

5.  One  sixth  of  tioo  apples  how  much  and  how  written? 

6.  One  ninth  of  twenty  oi'anges,  how  much,  and  how 
written  1 

Is  it  a  proper  or  improper  fraction  ? 

7.  OxiQ  sixth  oi' four  bushels  how  much?  how  written  ? 
is  it  a  proper,  or  improper  fraction  ? 

8.  One  tenth  of  forty  bushels,  how  much?  how  written  ? 
is  it  a  proper  or  improper  fraction  ? 

9.  One  te7ith  of  three  oranges,  how  much  ?  how  express- 
ed ? 


ADDITION.  187 

10.  Three  tenths  of  three  oranges,  how  much  ?  how  ex- 
pressed. 

11.  Four  siir//w  oi  twelve  apples,  ho'W  mxxclil  how  ex- 
pressed  ? 

12.  Three  thousand  tenths  of  ihousandtlis. 

13.  Four  billions,  six  thousand,  and  five  ten  thousandtlis. 

14.  Sixteen   bilhons,   three    hundred  and  six  milHons, 
five  hundred  thousand,  and  six  tenths  of  millionths. 

15.  Five  triUion,    five  million,   five  units,  and  three 
hundred  and  sixty  five  milUontlis. 

It).  Sixteen  hundred  and  twenty  four,  and  iowx  tenths  of 
biUionths. 


ADDITION. 

Let  the  pupil  add  the  following  numbers  : 
1 

Two  hundred  and  six  million  ;  twenty  four  thousand,  five 
hundred  and  six. 

Thirty  seven  billion,  twenty  six  thousand  and  three. 

Four  hundred  and  seventy  nine  billion,  six  hundred  and 
sixty  seven  million,  nine  hundred  and  eighty  four  thou- 
sand,  six  hundred  and  ninety  nine. 

Fifteen  million,  seventy  seven  thousand,  nine  hundred. 

Thirty  six  trillion,  four  hundred  million,  and  six. 

Four  quadrillion,  seventeen  million,  three  hundred  and 
six. 

Six  quadrillion,  fourteen  trillion,  seventeen  million,  four- 
teen thousand,  three  hundred  and  nine. 

Twenty  four  sextillion,  five  hundred  million  and  nine. 
2 

Sixteen  thousand,  four  hundred  and  sixty  four,  and  nine 
tenths. 

Two  hundred  and  sixty  nine  million,  fourteen  hundred 
and  three,  and  thirteen  hundredths. 

Forty  four  million  three  thousand  and  six,  and  twenty 
thousandths. 

Five  hundred  million,  nine  hundred  and  ninety  nine 
thousand,  eight  hundred  and  seventy  nine,  and  two  hun- 
dred and  sixty  four  tenths  of  thousandths. 


188  ARITHMETIC.      PART   THIRD. 

Six  hundred  and  seventeen  thousand,  four  hundred  and 
sixty  eight,  and  five  hundred  and  seventy  nine  hundredths 
of  thousandths. 
Forty  six  million,  nine  thousand,  and  seventy  millionths. 

3 
Add  two  twelfths,  three  fourths,  and  four  sixths.    (See 
page  166.) 

4 
Add   twenty   four  fiftieths,  sixteen  tenths,  and  twenty 
halves. 

5 
Add  forty  nine  eightieths,  seventy  nine  fortieths,  and  two 
hundred  thousandths. 

6 
Add   nine   twenty  sevenths,  thirteen  forty  fourths,  and 
twenty  nine  seventieths. 


SUBTRACTION. 

1 

From, 

Three  hundred  and  sixty  nine  million,  four  hun- 
dred  twenty  seven  thousand,  three  hundred  seventy  six. 

Subtract, 

Two  hundred  and  ninety  three  million,  four  hun- 
dred and  eighty  three  thousand,  nine  hundred  and 
eighty  seven. 

2 

From, 

Twenty  four  billion  six  hundred  and  thirteen  mil- 
lion, four  hundred  and  forty  four  thousand,  eight  hundred 
and  eighty  six,  and  twenty  nine  hundredths. 

Subtract, 

Sixteen  billions,  twenty  four  thousand  and  sixteen,  and 
tour  hundred  and  six  thousandths. 

3 

From', 

Sixty  four  sextillion,  ninety  trillion,  seven  billion,  twen- 
ty nine  million,  forty  thousand  three  hundred  and  six,  and 
twenty  nine  tenths  of  millionths, 


MULTIPLICATION.  189 

Subtract, 

Fourteen  quintillions,  nine  quadrillions,  seven  trillions, 
fourteen  thousand  and  eighty,  and  seven  hundredths  of 
milliontJis. 

4 
From  nine  twelfths,  subtract  two  fifths.       (See  page 
,      166.) 

5 
From  thirteen  twenty  sevenths,  subtract  three  twenty 
fourths. 

6 
From  ihreefijth^,  subtract  twenty  nine  seventy  sevenths. 

7 
From, 

Twelve  hundred  and  six,  four  hicndred  and  twentieths, 
Subtract, 
Four  hundred  and  nine,  7ww  hundred  and  ninetieths. 


MULTIPLICATION. 

1.  Multiply  32694302  by  365. 

2.  Multiply  24,2  by  27  (See  page  108.) 

3.  Multiply  324,92  by  236. 

4.  Multiply  236,49  by  2,4. 

5.  Multii)ly  47,2935  by  2,68432. 

6.  Multiply  876,24  by  32,94. 

7.  Multiply  14  yds.  3  qrs.  2  na.  by  28. 

8.  Multiply  8  le.  2  m.  6  fur.  22  po.  by  362. 

9.  Multiply  2  bu.  3  pk.   1  qr.  1  pt.  by  172. 

10.  Multiply  I  by  3  (Seepage  112.) 

11.  Multiply  ^  bv  48. 

12.  Multiply  if  by  32. 

13.  Muhiply  \2  by  |  (Seepage  116.) 

14.  Multiply  24  by   f. 

15.  Multiply  324  by  /^. 

16.  Muhiply  2342  by  Jv*- 

17.  Multiply  f  by  I  (Se'e  page  123.) 

18.  Multiply  f  by  f. 

19.  Multiply  f  by  f 

20.  Multiply  j\  by  ||. 
•21.  Muhiply  IP  by  if 


190  ARITHMETIC.       THIBD    PART. 

SUMS    FOR    MENTAL    EXERCISE. 

Multiply  5  and  f  by  f . 

Let  such  sums  be  stated  thus: 

One  fourth  of  5  is  1  unit,  and  1  remains. 

This  remaining  1  is  changed  to  sixths  and  added  to  the 
§  making  f 

One  fourth  of  one  sixth  would  be  ^V)  therefore  one 
fourth  of  eight  sixth  is  J,, 

In  the  above  operation  we  find  that  one  fourth  of  5  is  1 
and  1  remains.  This  remainder  is  changed  to  sixths  and 
added  to  the  fraction  |,  and  then  is  divided  by  4.  The  an- 
swer  is  1  and  ;/y. 

1 .  If  a  yard  of  muslin  cost  2i,  what  will  \  a  yard  cost  ? 
What  is  i  of  2i  ? 

2.  If  a  barrel  of  wine  cost  lOi  dollars,  what  cost  i  a 
barrel?     What  is  i  of  lOi  ? 

3.  If  4  bushels  of  rye  cost  8  dollars  and  |,  what  cost  2 
bushels  ?     What  is  i  of  8f  ? 

4.  If  you  have  2i  oranges,  and  givei  away,  how  much 
do  you  keep  ?     What  is  |  of  2i  ?     What  is  ^  of  8i  ? 

5.  If  9  bushels  of  wheat  cost  ISf  dollars,  how  much  is 
that  a  bushel  ?     What  is  i  of  18|  ? 

6.  If  12  pieces  of  linen  cost  16f  dollars,  how  much  is 
that  by  the  piece  ? 

7.  If  8  gallons  of  brandy  cost  14|  dollars,  how  much 
is  that  a  gallon  ? 

8.  If  8  yards  of  broadcloth  cost  28|  dollars,  how  much 
is  that  a  yard  ? 

9.  How  much  would  4  yards  cost  ? 

10.  If  a  man  bought  8  barrels  of  cider  for  25f  dollars, 
how  much  is  one  barrel  ? 

1 1 .  How  much  is  9  barrels  ? 

12.  If  12  yards  of  linen  cambric  cost  42|  dollars,  what 
would  7  yards  cost  ? 

13.  If  you  have  12  dollars  and  f  and  lose  3i  times  as 
much,  how  much  do  you  lose  ? 

We  first  multiply  the  12|  by  3  and  then  by  |. 

14.  3  times  12  is  36  and  3  times  f  is  f  or  1,  which 
added  to  36  is  37. 

15.  12  and  f  multiplied  by  i  is  6  and  i  which  added  to 
the  37  makes  43  and  i. 

16.  Multiply  8  and  |  by  4  and  i. 


DIVISION. 


191 


In  doing  this  sum,  first  multiply  the  8  and  then  the  frac- 
tion by  4,  and  add  the  products  together.  Then  multi- 
ply the  8,  and  the  fraction  by  i,  and  add  these  to  the 
former  products. 

Thus  4  times  8  is  32.  Four  times  |  is  |,  which  is  1 
and  |.     This  added  to  32  is  33  and  |. 

One  third  of  8  is  2,  and  2  remains.  Add  2  to  the  33 
making  35.  Change  the  remainder  to  fifths  and  add  the 
f  making  '/.  One  third  of  onejijlh  would  be  y'j,  there- 
fore ^  of  '/  is  If,  which  added  to  35  and  |  makes  35  and 
f  1,  which  equals  36  and  ■^^. 

17.  Multiply  5  and  f  by  2  and  i. 

18.  Multiply  12  and  f  by  2  and  |. 

19.  Multiply  9  and  y\  by  6  and  f . 

20.  Multiply  7  arid  |  by  4  and  f ! 

21.  Multiply  11  and  f  by  3  and  4. 

22.  Multiply  8  and  ^  by  8  and  1. 

23.  Multiply  10  and  |  by  7  and  \. 

24.  If  you  buy  9  and  |  gallons  of  wine  and  return  2^ 
times  as  much,  how  much  do  you  return  ? 

25.  If  one  boy  takes  12  apples  and  }  and  another  takes 
5i  times  as  many,  how  many  does  the  last  take  ? 

26.  If  one  room  requires  12  and  ^-  yards  of  carpeting, 
and  another  requires  3  and  |  times  as  much,  how  much  is 
required  ? 


1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 
10. 
11. 


Div 
Div 
Div 
Div 
Div 
Div 
Div 
Div 
Div 
Div 
Div 


12.  Div 


DIVISION. 

de  9123648  bv  79632. 

de  246,2  bv  23  (See  page  152.) 

de  2394,609  by  235. 

de  3246,9214  by  39. 

de  32.4  by  9,4  (See  page  152.) 

de  3294  by  2,79. 

de  324,976  by  2,4  (See  page  152.) 

de  329,42  by  3,24. 

de  329021,4639  by  296,029. 

de  112£.  12s.  7d.  4qrs.  by  38. 

de  29  yds.  2  qrs.  3  na.  by  39. 

de  2  m.  5  fur.  17  po.  3  yds.  by  91. 


192  AKITHMETIC.       PART    THIRD. 

13.  Divide  12by|  (Seepage  137.) 

14.  Divide  128  by  f . 

15.  Divide  418  by  tV- 
IG.  Divide  324  by /g . 

17.  Divide  3297  by  if. 

18.  Divide  if  by  6  (See  page  140.) 

19.  Divide  f|  by  16. 

20.  Divide  |||-  by  27. 

21.  Divide  Ifixby  361. 

22.  Divide  ^iff ^  by  249. 

23.  Divide  |  by"^  (See  page  144.) 

24.  Divide  j%  by  f 
2.5.  Divide  ^\  by  f. 
26.  Divide  III  by  iif 


EXAMPLES    FOR   MENTAL    EXERCISE. 

1.  Divide  i  by  |.     Divide  ^by^L  (See  page  164.) 

2.  Divide  i  by  -J^.     Divide  i"by  /_. 

3.  Divide  f  by  y\.     Divide  i  by  J  . 

4.  Divide  |  by  ^.     Divide  f  by  Jg. 

5.  Divide  f  by  j\.     Divide  j\  by  j. 

6.  Divive  j%  by  f .     Divide  f  by  f_. 

7.  How  many  times  is  i  contained  in  y"^  ? 

8.  How  many  times  is  |  contained  in  |  ? 

9.  How  many  times  is  ^  contained  in  -^  ? 

10.  How  many  times  is  f  contained  in  |  ? 

11.  If  beef  is  1  of  a  dollar  a  pound,  how  much  can  be 
bought  for  1  of  a  dollar. 

12.  If  a  yard  of  muslin  cost  J^  of  a  dollar,  how  much 
can  be  bought  for  |^  of  a  dollar  ? 

In  case  the  divisor  and  dividend  have  whole  numbers 
with  the  fractions,  the  whole  numbers  must  be  7-educed  also, 
with  the  fractions,  to  a  common  denominator. 

Thus  if  we  wish  to  find  how  many  |  there  are  in  4  and 
^,  we  must  change  the  4  and  |  to  twelfths,  and  the  §  to 
twelfths  also,  and  then  divide  as  before.  Tiius;  4  and  | 
is  f  |,  and  |  is  j\. 

In  57  twelths,  there  are  7  times  8  twelfths,  and  one 
twelfth  left  over.      This  one  twelfth,  is  one  eighth  of  the 


divisor  ^j. 


KEDUCTION.  193 

The  answer  then  is  7  and  }.  That  is,  4a  contains  f , 
just  7  times  and  i  of  another  time. 

Again  ;  how  often  is  2|  contained  in  5|  ?  First,  reduce 
the  divisor  and  dividend  to  fractions  of  a  common  deno- 
minator. 

2f  is  f |,  and  5f  is  f|. 

Divide  69  twelfths  by  32  twelfths,  and  the  answer  is  2 
and  5  twelfths  left  over. 

This  5  twelfths  is  5  thirty  secondths  of  the  divisor.  For 
Jive  twelfths  is  3%  of  32  twelfths. 

1.  How  many  times  is  1|  contained  in  8|? 

2.  How  many  times  is  2^  contained  in  5a  ? 

3.  How  many  times  is  9|  contained  in  16a  ] 

4.  If  you  distribute  13|  lbs.  of  flour  among  a  certain 
number  of  persons,  and  give  2|  lbs.  to  each,  to  how  many 
persons  do  you  give  ? 

5.  If  4|  bushels  of  wheat  last  a  family  one  week,  how 
long  will  12a  bushels  last  them? 

6.  If  55  tons  of  hay  will  keep  a  horse  6  months,  how 
many  horses  will  12f  tons  keep  during  the  same  time  ? 

7.  If  a  cistern  is  filled  in  3f  of  an  hour,  how  many  times 
will  the  cistern  be  filled  in  12|  hours? 

8.  If  you  distribute  ISf  dollars  among  the  poor,  and 
give  2A  dollars  to  each  person,  to  how  many  do  you  give  ? 

9.  At  3^  dollars  a  lb.  how  many  pounds  of  gum  can  be 
bought  for  24f  dollars  1 

10.  How  many  times  is  |  contained  in  2a  ? 

11.  How  many  times  is  5|  contained  in  8^  ? 

12.  How  many  times  is  2|  contained  in  14|  ? 

13.  How  many  times  is  3|  contained  in  7|? 

14.  How  many  times  is  5f  contained  in  12|  ? 


REDUCTION.       " 

1.  In  29  gallons  how  many  quarts  ?     (See  page  157.) 

2.  In  65  pints  how  many  gallons  ? 

3.  In  2j£,  14s.  9d.  3qrs.  how  many  farthings  ? 

4.  In  923469  farthings,  how  many  pounds,  shillings, 
and  pence  ? 

5.  Reduce  ^'to  a  decimal.     (See  page  162.) 

17 


194  ARITHMETIC.       PART   THIRD. 

6.  Reduce  |i  to  a  decimal. 

7.  Reduce  ^|f  to  a  decimal. 

8.  Reduce  1 1  and  j\  to  a  common  denominator.  (See 
page  166.) 

9.  Reduce  /^  ^3  -^j  to  a  common  denominator. 

10.  Reduce  -fj  -^-^  -^-^  to  a  common  denominator. 

11.  Reduce  -^f-^^  to  its  lowest  terms.     (Seepage  169.) 

12.  Reduce  |m  to  its  lowest  terms. 

13.  Reduce  ||  to  its  lowest  terms. 

14.  Reduce  |  of  a  guinea  to  the  fraction  of  a  pound. 
(See  page  171.) 

15.  Reduce  X9I46  to  the  fraction  of  a  foot. 

16.  Reduce  |  of  f  of  f  of  a  pound  to  the  fraction  of  a 
shilling. 

17.  Reduce  |  of  f  of  3  shillings  to  the   fraction  of  a 
pound. 

18.  What  is  the  value  of  |  of  a  ton  in  lbs  ?     (See  page 
172.) 

19.  How  many  ounces  in   |  of  a    lb.   Apothecary's 
weight  ? 

20.  How  many  pints  in  -^^  of  a  bushel  1 

21.  Reduce  8  oz.  6  pwts.  to  the  fraction  of  a  lb.  Troy. 
(See  page  173.) 

22.  Reduce  4  days  16  hours  to  the  fraction  of  a  year. 

23.  Reduce  36  gals.  4  qts.  to  the   decimal  of  a  hogs- 
head.     (See  page  174.) 

24.  Reduce  lid.  3qrs.  to  the  decimal  of  a  shilling. 
What  is  the  value  of  ,169432  of  a  ton  ?  (See  page  176.) 

25.  What  is  the  value  of  ,24694  of  a  £  ? 
What  is  the  value  of  ,396  of  an  hour  ? 

26.  Reduce  7s.  8d.  of  each  of  the  different  currencies 
to  the  same  value  in  Federal  money.     (See  page  179.) 

27.  Reduce  $6,  29  to  the  same  value  in  each  of  the 
different  currencies.    (See  page  180.) 


INTEREST. 

In  conducting  business,  men  often  find  it  necessary  to 
borrow  money  of  each  other,  and  it  is  customary  to  pay 
those  who  lend,  for  the  use  of  their  money  until  it  is  re- 
turned. 


INTEREST.  195 

The  sum  of  money  lent,  is  csWed  the  principal. 

The  sum  paidjor  the  use  of  money,  is  called  interest. 

Amount  is  the  principal  and  interest  added  together. 

Per  annum  signifies  hy  the  year. 

It  is  customary  to  pay  a  certain  sum  for  every  hundred 
dollars,  pounds,  &c.  Thus  in  New  England  six  dollars  a 
year  is  paid  for  the  use  of  every  hundred,  and  in  New 
York  seven  dollars  for  every  hundred  that  is  borrowed. 
The  expressions  six  per  cent.,  seven  per  cent  ,  &c.  signify 
that  six  or  seven  dollars  are  paid  for  every  hundred  bor- 
rowed.  Per  signifies  for  and  cent,  is  the  abreviation  of 
centum,  the  Latin  word  for  hundred.  Rate  per  cent.,  then, 
signifies  rale  hy  the  hundred.  When  a  man  borrows  a 
sum  of  money  he  gives  to  the  one  of  whom  ho  borrows  a 
writing  in  this  form  : 
.$500  ,00.  Hartford,  April  1,  1832. 

On  demand  I  promise  to  pay  D.  F.  Robinson  or  order, 
five  hundred  dollars  with  interest,  value  received. 

Samuel  Jones. 

This  is  called  a  note  and  is  said  to  he  on  interest. 

In  this  case  the  borrower,  Samuel  Jones,  is  obligated  to 
pay  six  dollars  a  year  for  each  hundred  dollars,  till  the 
$500  are  returned. 

In  Connecticut,  the  law  does  not  permit  men  to  receive 
any  more  than  6  per  cent,  interest ;  in  New  York  it  allows 
7  per  cent.,  and  the  rate  by  law  varies  in  the  different 
states.  When  the  rate  per  cent,  is  not  mentioned,  it  is 
always  to  be  understood  that  the  interest  is  what  is  allowed 
by  the  laws  of  the  state  where  the  note  is  given. 

Usury  is  taking  more  interest  than  the  law  allows. 

Legal  interest  \s  the  rate  allowed  by  law. 

In  all  notes  on  interest,  if  no  particular  r«fef)er  cew/. 
is  mentioned,  it  is  always  understood  to  be  legal  interest 
that  is  promised.  In  this  work  6  per  cent,  will  be  under- 
stood when  no  rate  per  cent,  is  mentioned. 

Sometimes  it  occurs  that  when  a  man  has  borrowed  a 
sum  of  money,  after  a  time  he  wishes  to  pay  a  part  of  the 
debt. 

In  this  case,  when  the  payment  is  made,  the  note  which 
was  given  to  the  lender  is  taken,  and  an  endorsement  is 
written  on  it,  stating  that  such  a  part  of  the  note  was  paid 


196  INTEREST. 

at  a  particular  time.  After  this  the  borrower  only  pays 
interest  for  that  part  of  the  debt  which  remains  unpaid. 

Notes  are  given  either  with  or  without  interest.  If  the 
words  "  with  interest"  are  not  written,  a  note  is  under- 
stood to  be  without  interest.  If  a  note  is  given  without  in- 
terest, promising  to  pay  at  a  certain  time,  after  that  time 
has  expired,  the  note  draws  interest  from  that  time. 

Notes  are  given  sometimes,  promising  to  pay  the  inter- 
est annually,  but  oftener  the  interest  is  not  to  be  paid  until 
the  note  is  paid. 

When  interest  is  paid  only  upon  the  sum  lent,  it  is  call- 
ed simple  interest. 

But  when  the  yearly  interest  is  added  each  year  to  the 
principal,  and  then  interest  is  taken  upon  both  principal  and 
interest,  it  is  called  compound  interest. 

The  laws  of  the  several  states  forbid  taking  compound 
interest ;  but  a  man  who  has  lent  money,  can  collect  the  in- 
terest every  year,  and  put  it  out  at  interest,  and  thus  gain 
compound  interest. 

But  when  a  man  borrows,  if  the  creditor  does  not  collect 
the  interest  every  year,  he  cannot  be  compelled  to  pay  in- 
terest on  the  interest. 

In  calculating  interest,  the  rote  per  cent,  is  a  certain 
number  of  hundredths  of  the  sum  lent.  Thus  if  1  per  cent, 
is  paid  for  $100,  it  is  yi^^  part  of  the  sum  lent.  If  6  per 
cent,  is  paid,  it  is  the  j^^  part  of  the  sum  lent. 

For  this  reason  all  calculations  in  interest  are  sums  in 
decimal  multiplication.  We  divide  by  the  denominator  to 
find  one  hundredth,  by  means  of  the  separatrix,  and  multi- 
ply by  the  numerator  to  find  the  required  number  of  hun- 
dredths. For  example,  if  we  wish  to  find  the  interest  of 
$263  for  one  year,  at  6  per  cent,  we  must  obtain  the  j-f^ 
part  of  the  $263.  This  is  done  by  dividing  by  the  denomi- 
nator 100,  by  means  of  a  separatrix,  and  multiplying  by 
the  numerator  6.  In  this  case  the  multiplication  is  done 
first. 


6 

$15,78 


INTEREST.  197 

The  rate  per  cent,  therefore,  may  always  be  written  as 
a  decimal  fraction  of  the  order  of  hundredths. 

1  per  cent,  is  written  ,01 

2  per  cent.  "  ,02 
J.  per  cent.  "  ,005 
i  per  cent.  "  ,0025 
I  per  cent.         "     ,0075 

Write  21  per  cent,  as  a  decimal  fraction. 

2  per  cent,  is  ,02,  and  i  per  cent,  is  ,005.     Ans.  ,025. 

Write  4  per  cent,  as  a  decimal  fraction.  44  per 

cent.  4|  per  cent. 5  per  cent. 7^  per 

cent.    — —  8  per  cent.    83  per  cent.    9  per 

cent.   9i  per  cent.  10  per  cent.  (10  per  cent. 

is  y'/o ;  decimally,  ,10.)  10^  per  cent.  11  per 

cent.  121  per  cent.  15  per  cent. 

1.  If  the  interest  on  ^1,  for  1  year,  be  6  cents,  what  will 
be  the  interest  on  $  17  for  the  same  time  ? 

It  will  be  17  times  6  cents,  or  6  times  17,  which  is  the 
same  thing : — 
$17 
,06 


1,02  Answer ;  that  is,  1  dollar  and  2  cents. 
To  find  the  interest  on  any  sum  for  1  year,  it  is  evident 
we  need  only  to  multiply  it  by  the  rate  per  cent,  written  as  a 
decimal  fraction.      The  product  will  be  the  interest  re- 
quired. 

What  is  the  interest  of  $121  at  3|  per' cent.  ?  at2i  per 
cent.  ?  at  8i  per  cent.  ?  at  9|  per  cent.  ?  at  4i  per  cent.  ? 
When  we  wish  to  obtain  the  interest  for  several  years, 
we  have  only  to  multiply  the  interest  of  one  year  by  the  num- 
ber of  years. 

Examples. 
What  is  the  interest  of  $214  for  4  years  at  2i  per  cent  ? 
for  3  yrs.  ?  for  9  yrs.  ?  for  24  yrs.  ? 

What  is  the  interest  of  $364,41  for  8  yrs.  at  6i  per  ct.  ? 
What  is  the  interest  sf  $1000  for  120  yrs.  ? 

Ans.    $7200. 
It  may  often  be  needful  to  calculate  the  interest  on  a 
sum,  for  a  less  time  than  a  year. 
17* 


198  INTEREST. 

When  this  is  needful  the  following  mode  is  the  most 
simple  and  expeditious. 

Let  the  interest  be  at  6  per  cent,  as  that  1^  the  most 
common  rate. 

At  6  per  cent,  each  dollar  gains  6  cents  a  year,  (or  12 
mo.)  6  cents  for  12  mo.  is  i  a  cent  (or  5  mills)  for  1  month. 

As  30  days  is  called  a  month,  in  calculating  interest,  5 
mills  a  month,  is  1  mill  for  every  6  days. 

Interest  at  6  per  cent  then  gains  on  each  dollar, 
$,06  a  year 
$,005  a  month 

$,001  for  every  6  days,  and  i  of  a  mill  for  each  day. 

Whenever  therefore  we  wish  to  calculate  the  interest  of 
any  sum  for  less  than  a  year,  we  can  first  calculate  the  in- 
tcrest  on  o7ie  dollar  for  the  given  time,  calculating  5  mills 
for  every  month,  1  mill  for  every  6  days,  and  ^  of  a  mill  for 
each  odd  day. 

After  finding  the  interest  for  one  dollar  we  can  multiply 
this  interest  by  the  number  of  dollars  in  the  sum. 

Examples. 

What  is  the  interest  of  $36  at  6  per  cent  for  9  mo.  12 
days?  for  6  mo.  3  days  ?  for  8  mo.  18  days  ? 

Note.  The  fractions  of  a  mill  had  better  be  changed  to 
decimals.  Thus  instead  of  writing  5i  mills  we  can  write 
j0055—  5|  mills  can  be  written  ,0053+.  (The  sign  of  ad- 
dition  is  added  to  the  last  because  there  are  more  decimal 
orders  that  may  be  added.) 

What  is  the  interest  of  $334  for  4  mo.  2  d.  ?  for  9  mo. 
5  d.  ?  for  7  mo.  4  d.  ? 

What  is  the  interest  of  $826  for  2  d.  ?  for  5  mo.  3  d.  ? 
for  10  d.  ?  for  9  mo.  16  d.  ? 

If  it  is  wished  to  obtain  the  interest  of  any  sum  for  less 
than  a  year,  at  any  other  than  6  per  cent,  the  method  is,  to 
find  the  interest  at  6  per  cent,  and  then  take  such  parts  of 
it,  as  the  rate  mentioned,  is  parts  of  6  per  cent. 

Thus  if  we  wish  to  find  the  interest  of  ^560  for  4  mo. 
8  d.  at  5  per  cent,  we  first  find  the  interest  at  6  per  cent. 


INTEREST.  199 

for  that  time,  and  then  subtract  i  of  the  sum  from  itself. 
.  For  the  interest  at  5  per  cent  is  ^  less  than  the  interest  at 
i^  per  cent. 

Thus  if  the  rate  is  3  percent,  we  must  take  i  of  the  in- 
terest at  6  per  cent. 

If  it  is  4  per  cent,  we  must  take  ^  (or  |)  of  the  interest 
at  6  per  cent,,&c. 

What  is  the  interest  of  $241,62  cents  for  8mo.  6d.  at  2 
per  cent.  ?  at  3  per  cent.  ?  at  4  per  cent.  ?  at  9  per  cent.  ? 
at  12i  per  cent  ?  at  15  per  cent.  ? 

What  is  the  interest  of  $54.81  for  18mo.  at  5  per  cent.  ? 

Ans.  $4.11. 

What  is  the  interest  of  $500  for  9mo.  9d.  at  8  per  cent.  ? 

Ans.  31.00. 

What  is  the  interest  of  $62.12  for  Imo.  20d.  at  4  per 
cent.  ?  Ans.  $0,345. 

What  is  the  interest  of  $85  for  lOmo.  15d.  at  121  per 
cent.  ?  Ans.  $9,295. 

Rules  for  calculating  Interest. 

To  find  the  interest  for  years. 

Multiply  the  sum  by  the  rate  per  cent,  as  a  decimal  of  the 
order  of  hundredths,  and  the  interest  for  one  year  is  found. 
Multiply  this  answer  by  the  number  of  years. 

To  find  the  Interest  for  months  and  days. 

Calculate  the  interest  on  one  dollar  for  the  given  time, 
thus  ;  calculate  5  7nills  for  every  month,  1  mill  for  every 
six  days,  and  }  of  a  mill  for  each  odd  day.  Add  these  to. 
getlier  and  multiply  the  answer  by  the  7iumber  of  dollars  and 
cents  in  the  sum,  pointing  off"  decimals  according  to  rule. 

If  the  rate  is  any  other  than  6  per  cent.,  calculate  the  in. 
terest  at  6  per  cent.,  and  then  add  to,  or  subtract  from  the 
sum  such  parts  of  itself,  as  the  rate  per  cent,  is  parts  of  6 
per  cent. 


200  arithmetic.    part  third. 

Examples. 
What  is  the  interest  of  $116,08  for  llmo.  I9d.  ? 

Ans.  $  6,422 

of  $200  for  8mo.  4d.  ?  8,132 

of    0,85  for  19mo.?  0,08 

of    8,50  for  lyr.  9mo.  12d.  ?  0,909 

of    675  for  Imo.  21d.  ?  5,737 

of    8673  for  lOd.?  14,455 

of    0,73  for  lOmo.  ?  0,36 


Rule  for  Sterling  Money. 

When  the  principal  is  pounds,  shillings,  and  pence,  reduce 
the  sum  to  the  decimal  of  a  £,  (see  page  174),  and  proceed 
as  in  federal  money.  The  answer  is  in  decimals  of  a  £, 
and  must  he  changed  hack  to  units  (see  page  176). 

What  is  the  interest  of  £36  ;  9s.  6id.  for  lyr.  ? 

Ans.  £2.3s.  9id. 
What  is  the  interest  of  £36  ;  10s.  for  18mo.  20d.  ? 

Ans.  £3.8s.  lid. 
What  is  the  interest  of  £95  for  9mo.  ?  Ans.  £4.5s.  6d. 
Find  the  interest  on  .£13  ;  3  ;  6  for  1  yr.  A.  15s.  9id. 
Find  the  interest  on  £13  ;  15s.  3id.  for  lyr.  6mo. 

A.  £1  ;  4  ;  9id. 
Find  the  interest  on  £75  ;  8  ;  4  for  5yrs.  2mo. 

A.  £23.7s.  7d. 
Find  the  interest  on  £174  ;  10  ;  6  for  3yrs.  6mo. 

A.  £36.13s. 
Find  the  interest  on  £325  ;  12 ;  3  for  5yrs. 

A.  £97.13s.8d. 
Find  the  interest  on  £150  ;  16 ;  8  for  4yrs.  7mo. 

A.  £41. 9s.  7d. 

VARIOUS  EXERCISES  IN  INTEREST. 

To  FIND  THE  Principal,   when  the  Time,  Rate  and 
Amount  are  known. 

If  in  lyr.  4mo.  the  interest  and  principal  of  a  sum 
■emount  to  $61,02  what  is  the  principal  ? 

We  first  find  what  will  be  the  amount  of  a  dollar  with  its 


INTEREST.  201 

interest,  for  the  given  time.  This  amounts  to  $1,08.  Now 
as  every  dollar  in  the  original  sum  gained  8  cents  interest, 
there  were  as  many  dollars  as  there  are  $1.08  in  $61,02. 

Am.  $56,50. 
Rule. 

Find  the  interest  of  %\for  the  given  time  and  add  to  it. 
Divide  the  sum  given  by  this  mnount. 

Examples. 

What  principal  at  8  per  cent,  will  amount  to  $85,12  in 
lyr.  6  mo.?  Ans.  $76. 

What  principal  at  6  per  cent,  will  amount  to  $99,311 
in  llmo.  9d.  ?  Ans.  $94. 

To    FIND    THE    PbINCIPAL,    WHEN   THE    TiME,    RaTE     AND 

Interest  are  known. 

What  sum  put  at  interest  at  6  per  cent,  will  gain  ^10,50  ? 

One  dollar  put  at  interest  for  that  time,  would  gain  $,08 

and  therefore  it  requires  as  many  one  dollars  as  there  are 

$,08  in  $10,50.  Ans.  $131,25. 

Rule. 

Find  tht  interest  of  $1  for  the  given  rate  and  time.     Di- 

vide  the  interest  given  by  this,  and  the  quotient  is  the  principal. 

Examples. 

A  man  paid  $4,52  interest  at  the  rate  of  G  per  cent,  at 
the  end  of  lyr.  4mo.  ;  what  was  the  principal  1 

A.  $56,50. 

A  man  received  $20  for  interest  on  a  certain  notd  at 
the  end  of  lyr.  at  the  rate  of  6  per  cent.  ;  what  was  the 
principal  ?  Ans.  $333,333i. 

To  FIND  THE  Rate,  when  the   Principal,  Interest, 
and  Time  are  known. 

If  $3,78  is  paid  for  using  $54,  lyr.  6mo.  what  is  the 
rate  per  cent.  ? 

If  this  sum  were  at  interest  at  one  per  cent,  it  would 
produce  $.54. 

As  many  times  therefore  as  8,54  is  contained  in  $3,78 
so  much  more  than  1  per  cent,  is  the  rate. 


202  arithmetic.     third  part. 

Rule. 

Divide  the  given  interest  hy  what  would  he  the  interest  of 
the  same  sum  at  1  per  cent. 

If  $2,34  is  paid  for  the  use  of  ^468  for  Imo.  what  is  the 
rate  percent.  ?  Ans.  6  per  cent. 

At  g46,80  for  the  use  of  $520  for  2yrs.  what  is  it  per 
cent.  ?  Ans.  4i  per  cent. 

To  FIND  THE  Time,  when  the  Principal,  Rate  and 
Interest  are  known. 

What  is  the  time  required  to  gain  $3,78  on  ^36  at  7 
per  cent.  ? 

We  first  find  what  would  be  the  interest  on  that  sum  for 
one  year,  at  7  per  cent. 

This  would  be  ^2,52.     As  many  times  as  this  sum  is 
contained  in  the  interest  mentioned  in  the  sum,  so  much 
more  time  than  one  year  is  required. 
Rule. 

Divide  the  interest  given,  hy  the  interest  which  the  princi. 
pal  tvould  gain  rd  the  same  rate,  in  one  year. 

Paid  §20  for  the  use  of  S600  at  8  per  cent.  ,-  what  was 
the  time  ?  Ans,  5mo. 

.  Paid  $28,242  for  the  use  of  $217,25  at  4  per  cent ; 
what  was  the  time  ?  Ans.  3yrs.  3mo. 

ENDORSEMENTS. 

In  transacting  business,  it  is  often  necessary  to  calculate 
interest  upon  notes,  where  partial  payments  have  been 
made,  and  endorsed  upon  the  note.  For  example,  a  man 
borrows  $500,  and  gives  his  note  promising  to  repay  it 
with  interest. 

Two  years  after,  he  pays  $150,  and  has  it  endorsed. 
Then  two  years  after,  he  pays  $75,  and  has  it  endorsed. 
At  the  end  of  six  years  he  is  ready  to  settle  the  note,  and 
the  question  is  how  much  interest  he  shall  pay. 

There  are  different  modes  established  by  the  laws  of 
different  states  on  this  subject. 


INTEREST.  203 

The  three  following  are  the  most  common.     The  first 

is  the  one  which  formerly  was  most  commonly  used. 

First  Method. 

Find  the  amount  of  the  principal  for  the%>}iole  time. 

Find  the  amount  of  each  payment  to  the  time  of  settlement. 

Add  the  amounts  of  the  payments,  and  subtract  them  from 

the  amount  of  the  principal. 

Example. 
On  April  1st,  1825, 1  gave  a  note  to  A.  B,  promising  to 
pay  him  $300  for  value  rec'd.  and  interest  on  the  same  at 
e  per  cent,  till  settlement. 
Oct.  1,  1825,  I  paid  v^lOO. 
April  16,  1826,     paid  $50. 
Dec.  1,  1827,       paid  |120. 
What  do  I  owe  on  April  1st,  1828  ? 
$  cts,  m. 
300,00,0  principal  dated  April  1, 1825.         yrs.  mo.  da. 
54,00,0  interest  up  to  April  1st,  1828.  3.     0.     0. 


354,00,0  amount  of  principal. 

100,00,0  1st  payment,  Oct.  1,  1825. 
15,00,0  interest  up  to  April  1st,  1828. 

1 15,00,0  amount  of  1st  payment. 


2.     6.     0. 


50,00,0  2nd  payment,  April  J  6th,  1820. 
5,87,5  interest  up  to  April  1st,  1828.         1.  11.  15. 


55,87,5  amount  of  second  payment. 


120,00,0  3rd  payment,  Dec  1st,  1827. 
2,40,0  interest  up  to  April  1st,  1828.         0.    4.    0. 


122,40,0  amount  of  3rd  payment. 

55,87,5        "        2nd  payment. 

115,00,0        «'         1st  payment. 


293,27,5  total  amount  of  payments. 


204  ARITHMETIC.      THIRD   PART. 

354,00,0  amount  of  principal. 

293,27,5  total  amount  of  payments  subtracted. 


A.  60,73,3Bremains  due  April  1st,  1828. 


Rule  in  Massachusetts. 

Find  the  Amount  of  the  Principal  to  the  time  lohen  one 
payment,  or  several papnents  together,  exceed  the  interest  due. 
From  this  subtract  the  payments  and  the  remainder  toill  be  a 
new  Principal.     Proceed  thus  till  the  time  of  settlement. 

Examples. 

For  value  received  I  promise  to  pay  James  Lawrence 
gll6,666  with  interest. 

May  1st,  1822. 

^116,666.  John  Smith. 

On  this  note  were  the  following  endorsements. 

Dec.  25,  1822,  received  $16,666. 

July  10,1823,  "       $  1,666. 

Sept.  1,  1824,  "       $  5,000. 

June  14,  1825,         "       $33,333. 

April  15, 1826,         "       $62,000. 

What  was  due  August  3,  1827  ? 

Ans.  $23,775. 

The  first  principal  on  interest  from  May  1, 
1822,  $116,666 

Interest  to  Dec.  25, 1822,  time  of  the  first 
payment  (7  months  24  days),  4,549 

Amount,  $121,215 
Payment,  Dec.  25,  exceeding  interest  then  due,  16,666 

Remainder  for  a  new  principal,  104,549 

Interest  from  Dec.  25,  1822,  to  June  14, 
1825  (29  months,  19  days),  15,490 

Amount,  $120,039 


INTEREST.  205 


Payment.  July  10,  1823,  less  than  interest 
then  due,  $  1,666 

Payment,  Sept.  1,  1824,  less  than 
interest  then  due,  5,000 

Payment  June  14,  1825,  exceed, 
inw  interest  then  due,  33,333 


$39,999 


Remainder  for  a  new  principal  (June  14, 
1825),  80,040 

Interest  from  June  14,  1825,  to  April  15, 
1826  (10  months  1  day),  4,01.'> 

Amount,  $84,055 
Payment,  April  15,  1825,  exceeding  inte- 
rest then  due,  62,000 

Remainder  for  a  new  principal  (April  15, 
1826),  $22,055 

Interest  due  Aug.  3,  1827,  from  April  15, 
1826  (15  months  18  days),  1,720 

Balance  due  Aug.  3, 1827,  $23,775 

The  Rule  now  adopted  in  Connecticut,  is  founded  on 
the  principle  that  interest  is  to  be  paid  by  the  year,  so  that 
if  a  man  pays  before  a  year  is  ended,  he  receives  interest 
on  all  he  pays,  from  the  time  he  pays  it,  to  the  end  of  the 
year  when  the  interest  is  due. 

Rule  in  Connecticut. 

If  the  payment  he  made  at  the  end  of  a  year  or  more,  add 
the  interest  due  on  the  whole  sum,  at  this  time,  to  the  princi- 
pal, and  subtract  the  payment. 

Whenever  other  payments  are  made,  proceed  in  the  same 
manner,  calculating  interest  on  the  principal  from  the  time 
of  the  last  payment. 

If  payment  is  made  hefore  a  year  has  elapsed  {from  the 
date  of  the  note,  or  Jrom  the  last  payment),  find  the  amo^int 
of  the  principal  for  one  year.  Find  also  the  amount  of  the 
payment  from  the  time  of  payment  to  the  endof  the  year  when 

18 


206  ARITHMETIC.       PART    THIRD. 

the  interest  would  be  due,  and  subtract  the  latter  from  the 
former.  If  hotoever  a  year  extends  beyond  the  time  of  settle- 
ment,find  the  amount  up  to  that  time,  instead  of  for  a  year. 

If  any  remainder  after  subtraction,  he  greater  than  the 
preceding  principal,  then  the  preceding  principalis  to  he  con- 
tinued as  the  principal  for  the  succeeding  time  instead  of  the 
remainder,  and  the  difference  to  be  regarded  as  so  much 
unpaid  interest. 

Let  interest  on  the  following  note  be  calculated  by  the 
three  differeni  rules. 

A  note  for  820,000  is  given  July  1st,  1825. 

1st  payment,  January  1st,  1826,  $1400 

2d       do.  ■;     January  1st,  1827,  2000 

3d       do;^;}      Seplernber  1st,  1827,  5000 

Settlemerit?January  1st,  1829. 

What  is  due  on  the  note  ? 

Ansti'ers, 

By  the  common  rule,  .$14,90Jr*,00 

By  the  Massachusetts  rule,  15,212,96 

By  the  Connecticut  rule,  15,209,47 

Let  the  following  be  calciilated  by  the  Cormecticut  rule. 

,$1000,00  Hartford,  Jan.  4,  1826. 

On  demand  I  promise  to  pay  James  Lowell,  or  order, 
one  thousand  dollars  with  interest ;  value  received. 

Hiram  Simpson. 

On  this  note  were  the  following  endorsements. 
Feb.  19,  1627,  received  -$200.00 

June  29,  1828,         "  500.00 

Nov.  14,  1828,         "  260.00 

Dec.  29, 1881,         »  25.00 

What  is  the  balance,  June  14,  1832  ? 

Answer  $204.49. 

Find  the  balance  due  on  the  following  note  by  the  Mas- 
sachusetts rule. 

$500.00.  Hartford,  Feb.  1,  1820. 

Value  received  1  promise  to  pay  A.  B.  or  order  five 
hundred  dollars  with  interest.  Samuel  Jones. 


INTEKKf 

iT. 

2 

Endorsements, 

May  1,  1820,  received, 

$40.00 

Nov.  14,  1820,      " 

8.00 

April  1,  18-21, 

12.00 

r^ay  1,1821, 

30.00 

How  much  remains  Sept.  Ifl, 

1821? 

Ans.  $445.57 

207 


Find  the  balance  due  on  the  following  note  by  the  Con- 
necticut  rule. 

For  value  received   I  promise   to   pay  G.  B.  or  order, 
eight  hundred  and  .seventy-five  dollars,  with  interest. 
.^875.00.  Sa.uuel  Jo?fKS. 

Hartford,  Jan.  10,  182!. 

Endorsements. 

Aug.  10,  1824,  received  $260.00 

Dec.  16,  1825,        "  300. 00 

March  1,  1820,       "  50.00 

July  1,  1827,           "     '  150.00 

What  was  due  Sept.  1,  1828  ? 

Ans.  $474.95. 

The  three  rules  us'ed  above,  are  all  considered  as  objec- 
tionable. 

By  the  first  rule,  when  a  man  pays  a  part  of  his  debt, 
his  payments  are  not  applied  to  discharging  the  interest, 
but  entirely  to  le.ssening  the  principal.  By  this  rule,  if  a 
man  should  borrow  a  sum  and  promise  to  pay  it,  with  the 
interest,  in  twenty-five  years,  if  he  should  simply  pay  what 
would  be  the  yearly  interest,  and  have  it  endorsed,  at  the 
end  of  25  years  the  debt  would  be  entirely  extinguished. 
Whereas  if  he  should  wait  till  the  end  of  the  time  agreed 
upon,  he  would  have  to  pay  the  original  sum  borrowed, 
and  the  yearly  interest  upon  it  also. 

The  objection  to  the  other  two  rules  is,  that  the  man 
who  makes  payments  before  the  time  of  settlement,  actu- 
ally  is  obliged  to  pay  more  than  one  who  pays  nothing  be- 
fore  that  time.  Thus  the  most  punctual  man  is  obliged  to 
pay  more  than  the  negligent. 

Compound  Interest  is  the  only  method,  which  will  do 
exact  justice  to  both  creditor  and  debtor.     For  a  man  who 


208  ARITHMETIC.        PART   THIRD. 

lends  money  is  fairly  entitled  to  receive  interest  at  the  end 
of  each  year  ;  and  then  by  investing  the  interest  in  other 
stock,  he  can  obtain  compound  interest.  The  borrower, 
therefore,  who  detains  this  yearly  interest,  ought,  in  jus- 
tice, to  pay  what  the  creditor  could  gain,  if  the  debtor 
were  punctual. 

COMPOUND  INTEREST. 

Compound  Interest  is  an  allowance  made  for  the  use  of 
the  sum  lent,  and  also  for  the  use  of  the  interest  when  it  is 
not  paid. 

Rule. 

Calculate  the  Interest,  and  add  it  to  the  principal  at  tlie 
end  of  a  year.  Make  the  Amount  a  new  principal  for  the 
next  year,  with  which  proceed  as  before,  till  the  time  of  set- 
tlement. 

1.  What  is  the  compound  interest  of  $256  for  3  years, 
at  6  per  cent.  ? 

$256  given  sum,  or  first  principal. 
,6 


15,36  interest,       \  ^^  ^^  ^^^^^  together. 

256,00  prmcipal,    ^  '= 

271,36  amount,  or  principal  for  2d  year. 
.06 


16,2816  compound  interest,  2d  year,  >    added 
271,36       principal,  do.         \  together. 

287,6416  amount,  or  principal  for  3d  vear. 
,06 


17,25846  compound  interest,  3d  year,  >    added 
287,641       principal,  do.         \  together. 

304,899       amount. 

256  first  principal  subtracted. 

A    $4S,899       compound  interest  for  3  years. 


INTEREST. 


2Q9 


2.  At  6  per  cent,  what  will  be  the  compound  interest, 

and  what  the  amount,  of  f  1  for  2  years  ?     what  the 

amount  for  3  years  ?  for  4  years  ?    for  5  years  ? 

for  6  years  ?     for  seven  years  ?     for  8 

years  ?  Ans.  to  the  last,  $1,593+ 

It  is  plain  that  the  amount  of  $2  for  any  given  time, 
will  be  2  times  as  much  as  the  amount  of  f  I  ;  the  amount 
of  $3  will  be  3  times  as  much,  &c. 

Hence,  we  may  form  the  amounts  of  #1,  for  several 
years,  into  a  table  of  multipliers  for  finding  the  amount  of 
any  sum  for  the  same  time. 

TABLE, 

Showing  the  amount  of  $1,  or  1J£,  &;c.  for  any  number 
of  years,  not  exceeding  24,  at  the  rates  of  5  and  6  per 
cent,  compound  interest. 


Y'rs. 
1 

2 
3 
4 
5 


5  per  cent. 

1,05 

1,1025 

1,15762  + 

1,21550+ 

1,27628+ 

1,34009+ 

1,40710  + 

1,47745  + 

1,55132  + 

1,62889+ 

1,71033+ 


12i  1,79585  + 


6  per  cent. 

1,06 
1,1236 
1,19101  + 
1,26247  + 
1,33822  + 
1,41851  + 
1,50363  + 
1,59384  + 
1,68947+ 
1,79084  + 
1,89829  + 
2,01219  + 


Y'rs. I    5  per  cent,    i    6  per  cent. 

131,88564+2,13292+ 
14|l,97993+ 2,26090+ 
15:2,07892+2,39655+ 
162,18287+  2,54035+ 
172,29201+2,69277+ 
18^2,40661 +2,65433+ 
192,52695  3,02559+ 
202,65329+  3,20713+ 
212,78596+3,39956  + 
22  2,92526+13,60353  + 
233,07152+3,81974  + 
24  3,22509+14,04893+ 

will  be 


Note  1.     Four  decimals  in  the  above  numbers 
sufficiently  accurate  for  most  operations. 

Note  2.  When  there  are  months  and  days,  you  may 
first  find  the  amount  for  the  years,  and  on  that  amount  cast 
the  interest  for  the  months  and  days ;  this,  added  to  the 
amount,  will  give  the  answer. 

3.  What  is  the  amount  of  $000,50  for  20  years,  at  5  per 
cent,  compound  interest  ?     at  6  per  cent.  ? 

$1  at  5  per  cent.,  by  the  table,  is  f  2,65329  ;  therefore, 
2,65329X600,50=$  1593,30+  Ans.  at  5  per  cent.  ;  and 
3,20713x600,50=^925,881+  Ans.  at  6  per  cent. 
18* 


4 


210  ARITHMETIC.       PART  THIRD. 

4.  What  is  the  amount  of  $40,20  at  6  per  cent,  com- 
pound interest,  for  4  years  ?     for  10  years  ? 


for  18  years  ?     for  12  years  ?     for  3  years  and 

4  months  ?     for  24  years,  6  months,  and  18  days  ? 

Ans.  to  last,  $168,137 

DISCOUNT. 

Discount  is  a  deduction  made  from  a  debt,  for  paying  it 
before  it  is  due. 

If,  for  example,  I  owe  a  man  $300  two  years  hence, 
and  am  willing  to  pay  him  now,  I  ought  to  pay  only  that 
su7n,  which.,  with  its  interest,  would  in  two  years,  amount 
to  $300. 

The  question  then  is,  what  sum,  together  with  its  inte- 
rest at  6  per  cent.,  would,  in  two  years,  amount  to  $300  ? 

Such  operations  are  performed  by  the  rule  for  finding 
the  principal,  when  the  time,  rate,  and  amount  are  given 
(see  page  201). 

The  sum  which,  in  the  time  mentioned,  would,  by  the 
addition  of  its  interest,  amount  to  the  sum  which  is  due,  is 
called  the  present  worth. 

What  is  the  present  worth  of  $834,  payable  in  1  yr.  7  mo. 
6  days,  discounting  at  the  rate  of  7  per  cent.  1 

Ans.  $750 

What  is  the  discount  on  $321,63,  due  4  years  hence, 
at  G  per  cent.  ?  Ans.  $62,26 

What  principal,  at  8  per  cent.,  in  1  yr.  6  mo.  will  amount 
to  $85,12  ?  Ans.  $76 

What  principal,  at  6  per  cent,  in  1 1  mo.  9  d.  will  amount 
to  $99,311?  Ans.  $94 

How  much  ready  money  must  be  paid  for  a  note  of  ^18, 
due  15  months  hence,  discounting  at  the  rate  of  6  per 
cent.  ?  Ans.  $16,744 

STOCK,  INSURANCE,  COMMISSION,  LOSS  AND 
GAIN,  DUTIES. 

Stock  is  a  name  for  money  invested  in  banks,  in  trade, 
in  insurance  companies,  or  loaned  to  a  national  govern- 
ment, for  the  purpose  of  receiving  interest. 


STOCK,  INSURANCE,  &C.  211 

Persons  who  invest  money  thus,  are  called  stockholders. 

When  stockholders  can  sell  their  right  to  stock,  for  more 

than  they  paid,  it  is  said  that  stock  has  risen,  and  when 

they  cannot  sell  it  for  as  much  as  they  paid,  it  is  said  that 

stock  has  fallen. 

Stock  is  bought  and  sold  in  shares,  of  from  $50  to  ^100 
a  share. 

The  nominal  value  of  a  share  is  the  amount  paid,  when 
the  stock  was  first  created. 

The  real  value  is  the  sum  for  which  a  share  will 
sell. 

When  stock  sells  for  its  nominal  value,  it  is  said  to  be  at 
par. 

When  it  sells  for  more  than  its  nominal  value,  it  is  said 
to  be  above  par,  and  when  for  less  it  is  below  par. 

When  stock  is  above  par  it  is  said  to  be  so  much  per 
cent,  advance. 

An  Insurance  Company,  is  a  body  of  men,  who  in  re- 
turn for  a  certain  compensation,  promise  to  pay  for  the 
loss  of  property  insured. 

The  written  engagement  they  give,  is  called  a  Policy. 
The  sum  paid  to  them  for  insurance,  is  called  Premium. 
Commission,  is  a  certain  sum  paid  to  a  person  called  a 
correspondent,  agent,Jactor,  or  broker,  for  assisting  in  trans- 
acting  business. 

Loss  and  Gain  refer  to  what  is  made  or  lost,  by  mer- 
chants, in  their  business. 

The  calculations  relating  to  stock,  insurance,  commiS' 
sion,  loss  and  gain,  and  duties,  are  performed  by  the  rule 
for  calculating  interest,  when  the  time  is  one  year  ? 
Rule. 
Multiply  the  sum  given,  by  the  rate  per  cent,  as  a  deci- 
mal.    (See  page  199.) 

Examples. 
Stock.— 1.  What  is  the  value  of  $350.00  of  stock  at 
105  per  cent,  that  is,  at  5   per  cent,   advance  ? 

Ans.  $367.50 
The  rate  here  is  105  per  cent.=  105  hundredths.     The 
question,  then,  is,  what  is  105  hundredths  of  350  ;  or,  mul- 
tiply 350  by  1.05. 


'212  ARITHMETIC.       PART   THIRD. 

2.  What  is  the  value  of  35  hundred  dollar  shares  of 
stock,  at  I  per  cent,  advance?  Rate  1.0075 

Ans  $3,526.^5 

3.  At  I12i  per  cent,  what  must  I  pay  for  $7,564.00  of 
stock?  Rate"l.  125.  Ans.  8,509.50 

4.  What  is  the  value  of  $615.75  of  stock,  at  30  per 
cent,  advance  ?  Ans.  $800,475 

5.  What  is  the  value  of  $7,650.00  of  stock  at  1191  per 
cent.  ?  Ans.  $9,141.75 

6.  What  is  the  value  of  $1,500.00  of  stock  at  110  per 
cent.  ?  Ans.  $1,650.00 

7.  What  is  the  value  of  ^3700  bank  stock  at  95i  per 
cent.,  that  is  at  4|  per  cent,  below  par?      Ans.  $3,533.50 

Insurance. — 1.  What  premium  must  be  paid  for  the 
insurance  of  a  vessel  and  cargo,  valued  at  $123,425.00, 
at  151  per  cent.  ? 

151  per  cent. ==.155,  and  the  question  is,  what  is  .155 
of  123,4''i5.  Ans.  $19,130,875 

2.  What  must  I  pay  annually  for  the  insurance  of  a 
house  worth  $3,500.00,  at  If  per  cent.  ?         Ans.  $61.25 

3.  What  must  be  paid  for  the  insurance  of  property,  at 

6  per  cent.,  to  the  amount  of  $2,500.00  ?      Ans.  $150.00 

4.  What  insurance  must  be  paid  on  $375,000-00,  at  5 
per  cent.  ?  Ans.  $18,750.00 

5.  What  premium  must  be  annually  paid  for  the  insur- 
ance of  a  house  worth  $10,050.00,  at  3  per  cent.;  and  a 
store  worth  $15,875.00,  at  4  per  cent.  ;  and  out  houses 
worth  $3,846  00,  at  5  per  cent.  ? 

6.  What  premium  must  be  annually  paid  for  the  insur- 
ance  of  a  Factory  worth  $30,946.00,  at  10  per  cent.  ;  and 

7  dwelling  houses,  worth  875.00  each,  at  8  percent.  ;  and 
3  grist  mills,  worth  $1,930.00,  apiece,  at  7  per  cent.  ;  and 
one  storeing  house,  worth  $9,859.00,  at  6  per  cent.  ?  Also, 
what  is  the  average  rate  of  insurance  on  the  whole  ? 

7.  If  I  pay  $930.00  annually  for  insurance,  at  5  per 
cent.,  what  is  the  value  of  the  property  insured  ? 

Here  930  is  .05  of  the  answer  ;  930 -r-. 05=$  18,600  An. 

Profit  anu  Loss. — 1.  Sold  a  bale  of  goods  at  ^735.00, 
by  which  I  gain  at  the  rate  of  6  per  cent.  What  sum  do 
I  gain?  Ans.  $44.10 

2.  In  selling  50  hhds.  of  molasses,  at  38  dollars  a  hhd., 
I  gain  10  per  cent.     What  is  my  gain  ?         Ans.  $190.00 


DUTIES.  213 

3.  In  selling  25  bales  of  cloth,  each  containing  27 
pieces,  and  each  piece  50  yards,  a  merchant  gained  20 
per  cent,  on  the  cost,  which  was  10  dollars  a  yard.  What 
did  he  gain,  and  what  did  he  sell  the  whole  for  ? 

Ans.   Gain  $67,500.00.     Whole  $405,000.00 

4.  A  merchant  gained  at  the  rate  of  15  per  cent,  in 
selling  the  following  articles  :  6  hhds.  of  brandy  for  which 
he  paid  $1.50  per  gal.  ;  7  barrels  of  flour,  cost  11  dollars 
a  barrel  ;  2  quintals  of  fish,  cost  4  cents  a  pound;  16 
hhds.  of  molasses,  cost  56  cents  per  gal.  and  25  bis.  of 
sugar,  containing  each  175  lbs.,  cost  9  cents  per  lb.  What 
was  his  gain  on  the  whole,  and  what  did  he  receive  in  all  ? 

Commission. — 1.  If  my  agent  sells  goods  to  the  amount 
of  $2,317.46,  what  is  his  commission  at  3^  per  cent.  ? 

Ans.  $75.31745 

2.  What  commission  must  be  allowed  for  a  purchase  of 
goods  to  the  amount  of  $1,286.00,  at2j  per  cent. 

Ans.  $32.15 

3.  What  commission  shall  I  allow  my  correspondent  for 
buying  and  selling  on  my  account,  to  the  amount  of 
$2,836.23,  at  3  per  cent.  ? 

4.  A  merchant  paid  his  correspondent  $25.00  commis- 
sion on  sales  to  the  amount  of  $1,250.00.  At  what  per 
cent,  was  the  commission  ? 

He  paid  him  y||^=j'-^y|^^.02=2  per  cent.  Ans. 

Duties. — Duty  is  a  certain  sum  paid  to  government 
for  articles  imported. 

When  duty  is  at  a  certain  rate  on  tlie  value,  it  is  said  to 
be  ad  valorem,  in  distinction  from  duties  imposed  on  the 
quantity. 

An  Invoice  is  a  written  account  of  articles  sent  to  a  pur- 
chaser,  factor,  or  consignee. 

In  computing  duties,  ad  valorem,  (or  ad  val.  as  it  is 
commonly  written,)  it  is  usual  in  custom  houses  to  add  one 
tenth  to  the  invoice  value,  before  casting  the  duty.  This 
makes  the  real  duty  one  tenth  greater  than  i\\e  nominaMn. 
ty.  It  will  be  equally  well  to  make  the  rate  one  tenth 
greater,  instead  of  increasing  the  invoice. 

1.  Find  the  duty  on  a  quantity  of  tea,  of  which  the  in- 
voice  is  $215.17,  at  50  per  cent. 

Ans.  $11 8.3435=:$1 18.3434 


214  ARITHMETIC.       PART    THiUD. 

In  this  example  we  mav  add,  as  directed  above,  one 
tenth  of  215.17,  to  ai^/lT.  Thus,  215,17+2J  .517= 
236  687.  Then  236.687X50=^  118.3435.  Or  we  may 
add  to  the  rate  .50,  one  tenth  of  ifseif=.05  :  thus,  .50-f- 
.05=55.     Then,  215. 17 x.55=$l  18.3435,  as  before. 

2.  Find  the  duty  on  a  quantity  of  hemp  at  13i  per  cent., 
of  which  the  invoice  is  ^664.59.  Tlie  second  of  the 
above  modes  is  recommended.     Another  might  be  used, 

viv.  :  to  find,  first,  the  duty  on  the  invoice  at  the  given  . 
rate,  and  add  to  it  one-tenth  of  itself.  Thus,  654. -59  X  I 
13|=<5;88. 36965.  Ans.  $97.206615        ' 

3.  What  is  the  duty  on  a  quantity  of  books,  of  which  the 
invoice  is  $1,670.33,"  at  20  per  cent.  ?       Ans.  $367.4726 


EQUATION  OF  PAYMENTS. 

Equation  of  payiiKnts  is  a  method  of  finding  a  time  for 
paying  several  debts  due  at  differeiit  times,  all  at  once  ; 
H.nd  in  such  a  way  that  both  creditor  and  debtor  will  have 
the  same  value,  as  if  the  debts  were  paid  at  the  several 
times  promised. 

For  if  a  man  pays  a  debt  before  it  is  due,  the  creditor 
gains  ;  if  he  pays  it  after  it  is  due,  the  debtor  gains. 

In  how  many  months  will  $1  gain  as  much  at  interest 
as  ^8  will  gain  in  4  months  ?  Now  as  the  $1  is  8  times 
less  than  8,  it  will  require  8  times  more  time,  or  8x4=32 
months. 

In  how  many  months  will  the  interest  on  $9  equal  the 
interest  on  $1  for  40  months  ? 

Supposing  a  man  owes  me  $12  in  3  months,  $18  in  4 
months,  and  $20  in  9  months.  He  wishes  to  pay  the 
whole  at  once  ;  in  what  time  ought  he  to  pay  ? 

$12  for  3  month.s=$l  for  36  months. 

$•18  for  4  months=$l  for  72  months. 

$20  for  9  months=$l  for  180  months. 


$50  288  months. 

Now  it  appears  that  it  will  be  the  same  to  him  to  have 
$1  for  36,  for  72,  and  for  180  months,  as  it  would  to  have 
the  12,  the  18,  and  the  20  dollars  for  the  number  of 
months  specified. 


RATIO.  215 

He  might  therefore  keep  $1  just  288  months,  and  it 
would  be  the  same  as  keeping  the  $50  for  the  number  of 
months  specified.  But  as  the  whole  sum  of  money  lent 
was  'f  50,  he  may  keep  this  only  onefftieth  (j\)  of  the  time 
he  might  keep  $1.  Therefore  divide  the  288  months  by 
the  50,  and  the  answer  is  5||  months. 


Rule  for  finding  the  mean  time  of  several  payments. 

Multiply  each  sum  by  the  time  of  its  payment.  Divide  the 
sum  of  these  products  by  the  sum  of  the  payments,  and  the 
quotient  is  the  mean  time. 

A  man  is  to  receive  $500  in  2mo.  ;  $100  in  5  mo.  ; 
8300  in  4  mo.  If  it  is  paid  all  at  once,  at  what  time  should 
the  payment  be  made  1 

A  man  owes  me  $300,  to  be  paid  as  follows  ;  i  in  3 
months  ;  |  in  4  months  ;  and  the  rest  in  G  months  ;  what 
is  the  mean  time  for  payment  ?  Aus.  4^  months. 


RATIO. 

The  word  ratio  means  relation  ;  and  when  it  is  asked 
what  ratio  one  number  has  to  another,  it  means  in  what 
relation  does  one  number  stand  to  another. 

Thus,  when  we  say  the  ratio  of  1  to  2  is  i,  we  mean 
that  the  relation  in  which  1  stands  to  2  is  that  of  ojie  half 
to  a.  whole. 

Again,  the  ratio  of  3  to  4  is  |,  that  is,  3  is  3  of  4,  or 
stands  in  the  relation  of  |  to  the  4.  So  also  the  ratio  of 
4  to  3  is  A  ;  for  the  4  is  4  thirds  of  3,  and  stands  to  it 
therefore  in  the  relation  of  |. 

What  is  the  relation  of  11  to  12?  of  12  to  11? 

When  therefore  we  find  the  ratio  of  one  number  to  an- 
other,  we  find  what  part  of  one  number  another  is 

Then  the  ratio  of  4  to  6  is  |  ;  that  is,  4  is  4  sixths  of  0. 

The  ratio  of  one  number  to  another,  then  may  always 
be  expressed  by  a  fraction  in  which  the^r*^  number  (called 
the  antecedent)  is  put  for  numerator,  and  the  second  number 
(called  the  consequent)  is  put  for  denominator.  Thus  the 
ratio  of  8  to  4  is  |.  This  is  an  improper  fraction,  and, 
changed  to  whole  numbers,  is  2  units.     The  ratio  of  8  to 


216  ARITHMETIC.       PART   THIRD. 

4,  then,  is  2.     That  is,  8  is  twice  4,  or  stands  to  4  in  the 
relation  of  a  duplicate  or  double* 


PROPORTION. 

When  quantities  have  the  same  ratio,  they  are  said  to  be 
■proportional  to  each  other.  Thus  the  ratio  of  2  to  4  is  i, 
and  the  ratio  of  4  to  8  is  |  ;  that  is,  1  has  the  same  relation 
to  2,  that  4  has  to  8,  and  therefore  these  numbers  are  called 
proportionals.  Again,  4  is  the  same  portion  or  part  of  8, 
as  10  is  of  20,  and  therefore  these  numbers  are  called 
proportionals.  A  proportion,  then,  is  a  combination  of  equal 
ratios. 

Points  are  used  to  indicate  that  there  is  a  proportion 
between  numbers.  Thus  4:8::9:18  is  read  thus  ;  4  has 
the  same  ratio  to  8,  that  9  has  to  18.  Or  more  briefly,  4 
is  to  8,  as  9  to  18. 

It  will  always  be  found  to  be  the  case  in  proportionals, 
that  multiplying  the  two  antecedents  into  the  two  consequents, 
produce  the  same  product. 

Thus,         2  :  4  :  :  6  :  12 

Here  let  the  consequent  4  be  multiplied  into  the  ante- 
cedent  6,  and  the  product  is  24  ;  and  let  the  atecedent  2 
be  multiplied  into  the  consequent  12,  and  the  product  also 
is  24. 

If  then  we  have  only  three  terms  in  a  proportion,  it  is 

*  The  pupil  needs  to  be  forewarned  that  there  is  a  d,ifference  be- 
tween French  and  English  mathematicians  in  expressing  Ratio. 

The  French  place  the  antecedent  as  denominator,  and  the  conse- 
quent as  numerator.  The  English,  on  the  contrary,  place  the  ante- 
cedent as  numerator,  and  the  consequent  as  denominator.  It  seems 
desirable  that  there  should  be  an  agreement  on  this  subject,  in  school 
books  at  least.  Two  of  the  most  popular  Arithmetics  now  in  use, 
have  adopted  the  French  method,  viz.  Colburn  and  Adams.  It  seems 
needful  to  mention  this,  that  pupils  may  not  be  needlessly  perplexed, 
if  called  upon  to  use  different  books. 

The  method  used  here,  is  the  English  ;  as  the  most  common,  and 
as  most  consonant  with  perspicuity  of  language.  For  there  seems 
to  be  no  propriety  in  saying  that  the  relation  of  2  to  4  is  A.  The 
ratio  between  these  two  numbers  maybe  either  |  or  |,  but  the  rela- 
tion of  2  to  4,  to  use  language  strictly  can  be  nothing  but  a. 


PROPORTION.  217 

easy  to  find  the  fourth.  For  when  we  have  multiplied  one 
antecedent  into  onn  consequent,  we  know  that  the  term 
left  out,  is  a  number  that,  multiplied  into  the  remaining 
term,  would  produce  the  same  product. 

Thus  let  one  term  be  left  out  of  this  proportion. 
8  :  4  :   :   12  : 

Here  the  consequent  is  gone  from  the  last  ratio.  We 
multiply  the  antecedent  12  into  the  consequent  4,  and  the 
answer  is  48.  We  now  know  that  the  term  left  out,  is  a 
number  which,  multiplied  into  the  8,  would  produce  48. 
This  number  is  found  by  dividing  48  by  8,  the  answer  is  6. 

Whenever,  therefore,  a  term  is  wanting  to  any  propor- 
tion, it  can  be  found  by  multiplying  one  of  the  antecedents 
by  one  of  the  consequents,  and  dividing  the  product  by 
the  remaining  number. 

What  is  the  number  left  out  in  this  proportion? 
3  :   12  :::  24  : 

What  is  the  number  left  out  in  this  proportion  ? 
9  :  8  :   :  27  : 

In  a  proportion,  the  two  middle  terms  are  called  the 
means,  and  the  first  and  last  terms  are  called  the  extremes. 

Rule  for  finding  a  fourth  term  in  a  Proportion. 

Multiply  the  means  together,  and  divide  the  product  by 
the  remaining  number. 

It  is  on  this  principle,  that  what  is  commonly  called  the 
"  Rule  of  Three,"  is  constructed.  By  this  process,  we 
find  a  fourth  term  when  three  terms  of  a  proportion  are 
given. 

Such  sums  as  the  following  are  done  by  this  rule. 

If  4  yards  of  broadcloth  cost  $12,  what  cost  9  yards  ? 

Now  the  cost  is  in  proportion  to  the  number  of  yards  ; 
that  is,  the  same  ratio  exists  between  the  number  of  yards, 
as  exists  between  the  cost  of  each. 

Thus, — as  4  yards  is  to  9  yards,  so  is  the  cost  of  4  yards 
to  the  cost  of  9  yards.  The  proportion,  then,  is  expressed 
thus  : 

yds.      yds.  $ 

4     :     9     :     :     12  : 

Here  the  term  wanting,  is  the  cost  of  9  yards  ;  and  if 
we  multiply  the  means  together,  and  divide  by  the  4,  the 
19 


218  ARITHMETIC.       PART  THIRD. 

answer  is  27  ;  which  is  the  other  term  of  the  proportion; 
and  is  the  cost  of  9  yards. 

Again,  if  a  family  of  10  persons  spend  3  bushels  of  malt 
a  week,  how  many  bushels  will  serve  at  the  same  rate 
when  the  family  consists  of  30  ? 

Now  there  is  the  same  ratio  between  the  number  of 
bushels  eaten,  as  between  the  numbers  in  the  family. 
That  is,  as  is  the  ratio  of  10  to  30,  so  is  the  ratio  of  3  to 
the  number  of  bushels  sought. 

Thus,  10      :     30  :   :  3  : 


Rule  of  Proportion  ;  or  Rule  of  Three. 

When  three  numbers  are  given,  place  that  one  as  third  term, 
which  is  of  the  same  kind  as  the  answer  sought.  If  the  an- 
swer is  to  be  greater  than  this  third  term,  place  the  greatest  of 
the  remaining  numbers  as  the  second  term,  and  the  less  num- 
ber as  first  term.  But  if  the  answer  is  to  he  less,  place  the 
less  number  as  second  term,  and  the  greater  as  first. 

In  either  case,  multiply  the  middle  and  third  terms  toge- 
ther, and  divide  the  product  by  the  first.  The  quotient  is  the 
answer,  and  is  always  of  the  same  order  as  the  third  term . 

Note.  This  rule  may  be  used  both  for  common,  com- 
pound,  and  decimal  numbers.  If  the  terms  are  compound, 
they  must  be  reduced  to  units  of  the  lowest  order  men- 
tioned. 

Many  of  the  sums  which  follow  will  be  better  understood 
if  performed  by  the  mode  of  analysis,  which  has  been 
explained  and  illustrated  in  a  former  part. 

For  example,  we  will  take  the  first  sum  done  by  the  rule 
of  proportion. 

If  4  yards  of  broadcloth  cost  $12,  what  cost  9  yards  ? 

We  reason  thus, — If  4  yards  cost  $12,  one  yard  must 
cost  di  fourth  of  $12.  Therefore,  divide  $12  by  4,  and  we 
have  the  cost  of  one  yard.  Multiply  this  by  9,  and  we 
have  the  cost  of  9  yards. 

(It  is  usually  best  to  multiply  first  and  then  divide,  and 
it  has  been  shown  that  this  is  more  convenient,  and  does 
not  alter  the  answer.) 


PROPORTION.  219 

Let  the  following  sums  be  done  by  the  Rule  of  Propor- 
tion, and  then  explained  by  analysis. 

1.  If  the  wages  of  15  weeks  come  to  64  dols.  19  cts.  what 
is  a  year's  wages  at  that  rate  ?       Ans.  $222,  52  cts.  5m. 

2.  A  man  bought  sheep  at  81.11  per  head,  to  the  amount 
of  $51. (J;  how  many  sheep  did  he  buy  ?  Aus.  46 

3.  Bought  4  pieces  of  cloth,  each  piece  containing  31 
yds.  at  1 6s.  6d.  per  yard,  (New  England  currency,)  what 
does  the  whole  amount  to  in  federal  money  ?      Ans.  6341 

When  a  tun  of  wine  cost  8140,  what  cost  a  quart  ? 

Ans.  13  cts.  8|-^m. 

4.  A  merchant  agreed  with  his  debtor,  that  if  he  would 
pay  him  down  65  cents  on  a  dollar,  he  would  give  him  up 
a  note  of  hand  of  219  dollars  88  cts.  I  demand  what  the 
debtor  must  pay  for  his  note  ?  Ans.  8162.42  cts,  2m. 

5.  If  12  horses  eat  30  bushels  of  oats  in  a  week,  how 
many  bushels  will  serve  45  horses  the  same  time  ? 

Ans.  1121  bushels. 

6.  Bought  a  piece  of  cloth  for  848.27  cts.  at  8 1.19  cts. 
per  yard  ;  how  many  yards  did  it  contain  ? 

Ans.  40  yds.  2  qrs.  y3_.o_ 

7.  Bought  3  hhds.  of  sugar,  each  weighing  8cwt.  Iqr. 
12  lb.  at  87.26  cts.  per  cwt. ;  what  come  they  to  ? 

Ans.  8182.1  ct.  S  m. 

8.  What  is  the  price  of  4  pieces  of  cloth,  the  first  piece 
containing  21,  the  second  23,  the  third  24,  and  the  fourth 
27  yards,  at  81.43  cts.  a  yard  ? 

Ans.  $135.85  cts.  21+23+24+27=95  yds. 

9.  Bought  3  hhds.  of  brandy,  containing  Qi,  62,  62i 
gals,  at  81  38  cts.  per  gallon.  I  demand  how  much  they  a- 
mount  to  ?  Ans.  8255.99  cts. 

10.  Suppose  agantleman's  income  is  ,^1836  a  year,  and 
he  spends  83.49  cts.  a  day, one  day  with  another,  how  much 
will  he  have  salved  at  the  year's  end?     Ans.  8562.15  cts. 

11.  A  merch't  bought  14  pipes  of  wine,  and  is  allowed  6 
months  credit,  but  for  ready  money  gets  it  8  cents  a  gallon 
cheaper  ;  how  much  did  he  save  by  paving  ready  money  ? 

'  Ans.  8141.12  cts. 

12.  Sold  a  ship  for  537Z.  and  I  owned  f  of  her  ;  what 
was  my  part  of  the  money?  Ans.  £201.7s.  6do 


220  ARITHMETIC.       PART  THIRD. 

13.  If  -f-g  of  a  ship  cost  $718.25  cents,  what  is  the  whole 
worth  ?  5  :  781,25  :  :  16  :  $2500  Ans. 

14.  If  I  buy  54  yards  of  cloth  for  £3 1.10s.  what  did  it 
cost  per  Ell  English  ?  Ans.  14s.  7d. 

15.  Bought  of  Mr.  Grocer  11  cwt.  3  qrs.  of  sugar,  at 
$8,12  per  cwt.  and  gave  him  James  Pay  well's  note  for 
£19. 7s.  (New  England  currency)  the  rest  I  pay  in  cash  ; 
tell  me  how  many  dollars  will  make  up  the  balance. 

Ans.  $30,91 

16.  If  a  staff  5  feet  long  casts  a  shade  on  level  ground  8 
feet,  what  is  the  height  of  that  steeple  whose  shade  at  the 
same  time  measures  181  feet  1  Ans.  I13i  ft. 

17.  If  a  gentleman  has  an  income  of  300  English  guineas 
a  year,  how  much  may  he  spend,  one  day  with  another,  to 
lay  up  500  dollars  at  the  year's  end  ?   Ans.  $2,46cts.  5m. 

18.  Bought  50  pieces  of  kerseys,  each  34  Ells  Flemish,  at 
8s.  4d.  per  Ell  English  ;  what  did  the  whole  cost  ?     £425 

19.  Bought  200  yards  of  cambric  for  £90,  but  being 
damaged,  I  am  willing  to  lose  £7.  10s.  by  the  sale  of  it ; 
what  must  I  demand  per  Ell  English  ?         Ans.  lOs.  3|d. 

20.  How  many  pieces  of  Holland,  each  20  Ells  Flemish, 
may  I  have  for  £23.8s.  at  6s.  6d.  per  Ell  English  ? 

x\ns.  6  pieces. 

2 1 .  A  merchant  bought  a  bale  of  cloth  containing  240  yds. 

at  the  rate  of  $7^  for  5  yards,  and  sold  it  again  at  the  rate 

of  $11^  for  7  yards  ;  did  he  gain  or  lose   by  the  bargain, 

and  how  much  ?  Ans.   He  gained  $25,71  cts.  4m. + 

22.  Bought  a  pipe  of  wine  for  84  dollars,  and  found  it  had 
leaked  out  12  gallons  ;  I  sold  the  remainder  at  121  cents 
ii  pint ;  what  did  I  gain  or  lose  ?  Ans.  I  gained  $30 

23.  A  gentleman  bought  18  pipes  of  wine  at  12s.  6d.  (N. 
Jersey  currency)  per  gallon  ;  how  many  dollars  will  pay 
the  purchase  ?  Ans.  $3780 

24.  Bought  a  quantity  of  plate,  weighing  15  lb.  11  oz.  13 
pwt.  17  gr.  how  many  dollars  will  pay  for  it,  at  the  rate  of 
12s.  7d.  (New  York  currency,)  per  ounce  ? 

Ans.  $301,50  cts.  2/_ni. 

25  A  factor  bought  a  certain  quantity  of  broadcloth  and 

drugget,  which  together  cost  £81    per  yard,  the  quantity 

of  broadcloth  was  50  yards,  at  18s.  per  yard,  and  for  every 

5  yards  of  broadcloth  he  had  9  yards  of  drugget ;   1  de. 


PKOPORTION.  821 

mand  how  many  yards  of  drugget  he  had,  and  what  it  cost 
him  per  yard  ?  Ans.  90  yards  at  8s.  per  yard. 

26.  If  I  give  1  eagle,  2  dollars,  8  dimes,  2  cents  and  5 
mills,  for  675  tops,  how  many  tops  will  19  mills  buy  ? 

Ans.  1  top. 

27.  If  100  dollars  gain  6  doUars  interest  in  a  year,  how 
much  will  49  dollars  gain  in  the  same  time  ? 

Ans.  $2,94  cts. 

28.  If  60  gallons  of  water,  in  one  hour,  fall  into  a  cistern 
containing  300  gallons,  and  by  a  pipe  in  the  cistern,  35  gaU 
Ions  run  out  in  an  hour  ;  in  what  time  will  it  be  filled  ? 

Ans.  in  12  hours. 

29.  A  and  B  depart  from  the  same  place  and  travel  the 
same  road  ;  but  A  goes  5  days  before  B,  at  the  rate  of  15 
miles  a  day  ;  B  follows  at  the  rate  of  20  miles  a  day  ; 
what  distance  must  he  travel  to  overtake  A  ? 

Ans.  300  miles. 


COMPOUND  PROPORTION. 

Compound  proportion,  is  a  method  of  performing  such 
operations  in  proportion,  as  require  two  or  more  stat- 
ings.  It  is  sometimes  called  Double  Rule  of  Three,  be- 
cause  its  operations  can  be  performed  by  two  operations  of 
the  Rule  of  Three. 

For  example  :  If  56  lbs.  of  bread  are  sufficient  for  7  men 
14  days,  how  much  bread  will  serve  21  men  3  days? 

Here  the  amount  of  bread  consumed  depends  upon  two 
circumstances,  the  number  of  days,  and  the  number  of  men. 

We  will  first  consider  the  quantity  of  bread  as  depend- 
ing upon  the  number  of  men,  supposing  the  number  of  days 
to  be  the  same. 

The  proportion  would  then  be  this  ; 

7  men  :  21  men  :  :  56  lbs.  to  the  number  of  lbs.  re- 
quired. 

Here  we  multiply  the  means  together,  and  divide  the 
answer  by  7,  and  the  answer  is  168.  That  is,  if  the  time 
was  the  same,  viz.  14  days,  the  21  men  would  eat  168  lbs, 
in  that  time. 

We  now  make  a  second  statement  thus  : 

19* 


,  222  ARITHMETIC.       PART  THIRD. 

H  days  :  3  days  :  :  168  lbs. :  number  of  lbs.  requir- 
ed. 

The  result  of  this  statement  is  36  lbs.  which  is  the  an- 
swer. 

In  performing  this  operation,  let  the  pupil  notice  that  in 
the  first  statement,  the  56  was  multiplied  by  the  21  and 
the  answer  divided  by  7.  This  gives  the  same  answer  as 
would  be  given,  did  we  divide  first,  and  then  multiply. 

That  is,  56  multiplied  by  21,  and  the  product  divided  by 
7,  is  the  same  as  56  divided  by  7  and  the  quotient  multi- 
plied  by  21. 

We  divide  by  7,  to  find  how  much  one  man  would  eat  in 
the  same  time,  or  14  days,  and  multiply  by  21,  to  find 
how  much  21  men  would  eat. 

When  we  make  the  second  statement,  as  we  have  found 
how  much  21  men  would  eat  in  14  days,  we  divided  the 
quantity  (168  lbs.)  by  14,  to  find  how  much  they  would 
eat  in  one  day,  and  then  multiply  by  3,  to  find  how  much 
they  would  eat  in  3  days.  But  in  this  case  also,  the  muU 
tiiMcation  is  AonQ  first. 

Let  the  pupil  also  notice  that  the  56  lbs,  was  multiplied 
by  21  and  divided  by  7,  and  then  that  the  answer  to  this 
(168  lbs.)  was  multiplied  by  3  and  divided  by  14.  Here 
21  and  3  are  used  as  multipliers,  and  14  and  7  are  used 
as  divisors. 

The  answer  will  be  the  same  (as  may  be  found  by  trial) 
if  56  is  multiplied  by  the  product  of  these  multipliers,  and 
the  answer  divided  by  the  product  of  the  divisors. 

It  is  on  this  principle  that  the  common  rule  in  compound 
proportion  is  constructed,  which  is  as  follows. 

Rule  of  Compound  Proportion. 

Make  the  number  which  is  of  the  same  kind  as  the  answer 
required,  the  third  term. 

Take  any  two  numbers  of  the  same  kind,  and  arrange 
them  in  regai-d  to  this  third  term,  according  to  the  rule  of 
proportion.  Then  take  any  other  two  numbers  of  the  same 
kind,  and  arrange  them  in  like  manner,  and  so  on  till  all  the 
numbers  are  used. 

Then  multiply  the  third  term,  by  the  product  of  the  second 


PROPORTION.  223 

terms,  and  divide  the  answer  by  the  product  of  the  first  terms. 
The  quotient  is  the  answer. 

Examples. 

1.  If  a  man  travel  273  miles  in  13  days,  travelling  only 
7  hours  a  day,  how  many  miles  will  he  travel  in  12  days  at 
the  rate  of  10  hours  a  day  ? 

Here  the  number,  which  is  of  the  same  kind  as  the 
answer  required,  is  the  273  miles,  and  this  is  put  as  third 
term. 

We  now  take  two  numbers  of  the  same  kind,  viz.  13 
days  and  12  days,  and  placing  them  according  to  the  rule 
of  simple  proportion,  the  question  would  stand  thus. 
13  :  12  :  :  273  : 

We  next  take  two  other  numbers  of  the  same  kind,  viz. 
10  hours,  and  7  hours,  and  arrange  them  under  the  former 
proportion  according  to  the  same  rule,  thus  : 
13  :  12 


7  :  10  ^    .  :  273 

We  now  multiply  the  273  by  the  product  of  12  and  10, 
and  divide  by  the  product  of  13  and  7  and  the  quotient  is 
the  answer. 

We  can  explain  this  process  analytically,  thus. 

We  divide  by  13,  to  find  how  much  the  man  would  tra- 
vel in  one  day,  at  the  rate  of  7  hours  per  day. 

We  multiply  by  the  12,  to  find  how  much  he  would 
travel  in  12  days,  at  the  same  rate. 

We  divide  by  7  to  find  how  much  he  would  travel  in  one 
hour,  and  multiply  by  10  to  find  how  much  he  would  trave  1 
in  10  hours. 

Let  the  pupils  explain  the  following  in  the  same  man- 
ner. 

Examples. 

2.  If  £100  in  one  year  gain  £5  interest,  what  will  be 
the  interest  of  £750  for  7  years  ?  Ans.  £262.  IDs. 

3.  What  principal  will  gain  £262. 10s.  in  7  years,  at  5 
per  cent,  per  annum  ?  Ans.  £750. 

4.  If  a  footman  travel  130  miles  in  3  days,  when  the 
days  are  12  hours  long  ;  in  how  many  days,  of  10  hours 
each,  may  he  travel  360  miles  ?  Ans.  9||  days. 


224  ARITHMETIC,   PART  THIRD. 

5.  If  120  bushels  of  corn  can  serve  14  horses  56  days, 
how  many  days  will  94  bushels  serve  6  horses  ? 

Ans.  102i|  days. 

6.  If  7  oz.  5  pwts.  of  bread  be  bought  at  4fd.  when  corn 
IS  at  4s.  2d.  per  bushel,  what  weight  of  it  may  be  bought 
for  is.  2d.  when  the  price  of  the  bushel  is  5s.  6d.  ? 

Ans.  lib.  4  oz.  3f|f  pwts. 

7.  If  the  carriage  of  13  cwt.  1  qr.  for  72  miles  be  £2. 
10s.  Gd.  what  will  be  the  carriage  of  7  cwt.  3  qrs.  for 
112  miles  ?  Ans.  £2.58.  lid.  1 //^q. 

H.  A  wall,  to  be  built  to  the  height  of  27  feet,  was  raised 
to  the  height  of  9  ft.  by  12  men  in  6  days  ;  how  many  men 
must  be  employed  to  finish  the  wall  in  4  days  at  the  same 
rate  of  working  ?  Ans.  36  men. 

9.  If  a  regiment  of  soldiers,  consisting  of  939  men,  can 
eat  351  quarters  of  wheat  in  7  months  ;  how  many  soldiers 
will  eat   1464  quarters  in  5  months,  at  that  rate  1 

Ans.  5483//^. 

10.  If  248  men,  in  5  days  of  11  hours  each,  dig  a  trench 
230  yards  long,  3  wide  and  2  deep ;  in  how  many  days 
of  9  hours  each,  will  24  men  dig  a  trench  of  420  yards 
long,  5  wide  and  3  deep  ?  Ans.  288^Vt- 

11.  If  6  men  build  a  wall  20  ft.  long,  6  ft.  high,  and  4  ft. 
thick,  in  16  days,  in  what  time  will  24  men  build  one  200 
ft.  long,  8  ft.  high,  and  6  ft.  thick  ?  Ans.  80  days. 

12.  If  the  freight  of  9  hhds.  of  sugar,  each  weighing  12 
cwt.,  20  leagues,  cost  £16,  what  must  be  paid  for  the 
freight  of  50  tierces,  each  weighing  2i  cwt.,  100  leagues  1 

Ans.  £921.  Is.  10|d. 

13.  If  4  reapers  receive  $11.04  for  3  days' work,  how 
many  men  may  be  hired  16  days  for  $103.04  ?  Ans.  7  men. 

14.  If  7  oz.  5  pwt.  of  bread  be  bought  for  4|d.  when 
corn  is  4s.  2d.  per  bushel,what  weight  of  it  may  be  bought 
for  Is.  2d.  when  the  price  per  bushel  is  5s.  6d.  ? 

Ans.  1  lb.  4  oz.  3||f  pwts. 

15.  If  8100  gain  $6  in  1  year,  what  will  400  gain  in 
9  months  ? 

16.  If  $100  gain  $6  in  1  year,  in  what  time  will  $400 
gain  $18  ? 

17.  If  $400  gain  $18  in  9  months,  what  is  the  rate  per 
cent,  per  annum  1 


FELLOWSHIP.  225 

18.  What  principal,  at  6  per  cent,  per  ami.,  will  gain  S 18 
in  9  months  ? 

19.  A  usurer  put  out  $75  at  interest,  and,  at  the  end  of  8 
months,  received,  for  principal  and  interest,  $79  ;  I  demand 
at  what  rate  per  cent,  he  received  interest.     Ans.  8  per  ct. 

20.  If  3  men  receive  JESy'y  for  19^  days  work,  how  much 
must  20  men  receive  for  lOOi  days'  ?       Ans.  £305  Os.  8d. 

21.  If  40  men  in  10  days,  can  reap  200  acres  of  grain, 
how  many  acres  can  14  men  reap  in  24  days  ? 

Ans.  168  acres. 

22.  If  14  men  in  24  days,  can  reap  108  acres  of  grain  ; 
how  many  acres  can  40  men  reap  in  10  days  ? 

Ans.  200  acres. 

23.  If  16  men  in  32  days,  can  mow  256  acres  of  grass  ;  in 
how  many  days  will  8  men  mow  96  acres  ?  Ans.  24  days. 

24.  If  4  men  mow  96  acres  in  12  days  ;  how  many  acres 
can  8  men  mow  in  16  days  ?  Ans.  256 

25.  If  a  family  of  16  persons  spend  8320  in  8  months  ; 
how  much  would  8  of  the  same  family  spend  in  24  months  ? 

Ans.  8480 
2(5.  If  a  family  of  8  persons  \n  24  months  spend  $480 ; 
how  much  would  they  spend,  if  their  number  were  doub- 
led, in  8  months  ?  Ans.  $320 
27.  If  12  men  build  a  wall  100  feet  long,  4  ft.  high,  and 
3  ft.  thick,  in  40  days ;  in  what  time  will  6  men  build  one, 
20  ft.  long,  6  ft.  high,  and  4  ft.  thick  ? 


FELLOWSHIP. 

The  Rule  of  Fellowship,  is  a  method  of  ascertaining  the 
respective  gains  or  losses  of  individuals  engaged  in  joint 
trade. 

Let  the  pupils  perform  the  following  sums  as  a  mental 
exercise. 

1.  Two  men  own  a  ticket;  the  first  owns  i,  and  the 
second  owns  |  of  it ;  the  ticket  draws  a  prize  of  40  dollars ; 
what  is  each  man's  share  of  the  money  ? 

2.  Two  men  purchase  a  ticket  for  4  dollars,  of  which 
one  man  pays  1  dollar,  and  the  other  3  dollars  ;  the  ticket 
draws  40  dollars  ;  what  is  each  man's  share  of  the  money  ? 


226  ARITHMETIC.       PART  THIRD. 

3.  A  and  B  bought  a  quantity  of  cotton  ;  A  paid  $100, 
and  B  $200  ;  they  sold  it  so  as  to  gain  ^30 ;  what  were 
theirrespective  shares  of  the  gain  ? 

The  value  of  v/hat  is  employed  in  trade  is  called  the 
Capital,  or  Stock.  The  gain  or  loss  to  be  shared  is  called 
the  Dividend. 

Each  man's  gain  or  loss  is  always  in  proportion  to  his 
share  of  the  stock,  and  on  this  principle  the  rule  is  made. 

Rule. 

As  the  whole  stock  is  to  each  man's  share  of  the  stock,  so 
is  the  whole  gain  or  loss,  to  his  share  of  the  gain  or  loss. 

4.  Two  persons  have  a  joint  stock  in  trade  ;  A  put  in 
^250,  and  B  $350  ;  they  gain  $400 ;  what  is  each  man's 
share  of  the  profit  ? 

Operation. 
A's  stock,     $250^  Then, 
B's  stock,     $350  (^  gQ^  .  250  : :  400  :  $166.666f  A's  gain. 

Whole  stock  $600  J  600  :  350  : :  400  :  $233.333|  B's  gain. 

The  pupil  will  perceive  that  the  procese  may  be  con- 
tracted  by  cutting  off  an  equal  number  of  ciphers  from  the 
frst  and  second,  or  frst  and  third  terms  ;  thus,  6  :  250  :  : 
4 :  166.666|,  &c. 

It  is  obvious  the  correctness  of  the  work  may  be  ascer- 
tained by  finding  whether  the  sums  of  the  shares  of  the 
gains  are  equal  to  the  whole  gain  ;  thus,  $166.666|4- 
$233.333i=$400,  whole  gain. 

5.  A,  B,  and  C,  trade  in  company  :  A's  capital  was  $175, 
B's  200,  and  C's  $500  ;  by  misfortune  they  lose  $250  ; 
what  loss  must  each  sustain  ?        (  $  50.,  A's  loss. 

Aws.  ■^$  57.142f     B's  loss. 
^$142,857^,     C's  loss. 

6.  Divide  600  among  3  persons,  so  that  their  shares  may 
be  to  each  other  as  1,  2,3,  respectively. 

Ans.  $100,  $200,  and  $300 

In  assessing  taxes,  it  is  customary  to  obtain  ^an  inven- 

tory  of  every  man's  property,  in  the  whole  town,  and  also 

a  list  of  the  number  of  polls.     Each  poll  is  rated  at  a  tax 


FELLOWSHIP. 


227 


of  a  certain  value.  From  the  whole  tax  to  be  raised  is 
taken  out  what  the  tax  on  polls  amounts  to,  and  the  re- 
mainder of  the  tax  is  to  be  assessed  on  the  property  in  the 
town. 

We  may  then  find  the  tax  upon  1  dollar,  and  make  a 
table  containing  the  taxes  on  1,2,  3,  &c.  to  10  dollars  ; 
then  on  20,  30,  &c.  to  100  dollars ;  and  then  on  100,  200, 
&c.  to  1000  dollars.  Then,  knowing  the  inventory  of 
any  mdividual,  it  is  easy  to  find  the  tax  upon  his  proper- 
ty. 

I.  A  certain  town,  valued  at  !$64530,  raises  a  tax  ol 
,^2259.90  ;  there  are  540  polls,  which  are  taxed  8,60 
each  ;  what  is  the  tax  on  a  dollar,  and  what  will  be  A's 
tax,  whose  real  estate  is  valued  at  ^1340,  hip  personal  pro- 
perty at  8874,  and  who  pays  for  2  polls  ? 

540  X  ,60  =  $324,    amount   of  the    noil    taxes,    and 
.$2259,90,— $324=1935,90,  to  be  assessed  on  propertv. 
$645301:  81935,90  : :  $1  :  ,03  ;  or,«|||^V=>03,  tax  on  S'l 
TABLE. 


dolls,    dolls. 

dolls,    dolls. 

dolls.       dolls 

Tax  on  1  is  ,03 

Tax  on 

10  is     ,30 

Tax 

on  100  is     3, 

2  "  ,06 

^^ 

20  "     ,60 

200  "     6, 

3  «  ,09 

30  "     ,90 

300  "     9, 

4  <'  ,12 

40  "  1,20 

400  "  12, 

5  "  ,15 

" 

50  "  1,50 

500  "   15, 

6  «  ,18 

(( 

60  «  1,80 

600  "  18, 

7  «  ,21 

70  «  2,10 

700  »  21, 

8  "  ,24 

80  "  2,40 

800  "  24, 

9  "  ,27 

90  "  2,70 

900  "  27, 
1000  »  30, 

Now,  to  find  A's  tax,  his  real  estate  being  $1340,  I 
find  by  the  table,  that 

The  tax  on        $1000        -        is        -        30, 
The  tax  on  300        -         -          -  9, 

The  tax  on  40        -         -  -  1,20 


Tax  on  his  real  estate  -         -          -     $40,20 

In  like  manner  I  find  the  tax  on  his  personal  >     ^g  oo 

property  to  be  ^         ' 

2  polls,  at  ,60  each,  are  1,20 

Amount,  $67,62 


228  ARITHMETIC.       PART    THIRD. 

2.  What  will  B's  tax  amount  to,  whose  inventory  is  874 
dollars  real,  and  210  dollars  personal  property,  and  who 
pays  for  3  polls  ?  Ans.  $34.32 

3.  What  will  be  the  tax  of  a  man,  paying  for  1  poll, 
whose  property  is  valued  at  $3482  ? at  ^768  ? 

Ans.  to  the  last,  $140.31 
Two  men  paid  10  dollars  for  the  use  of  a  pasture  1 
month  ;  A  kept  in  24  cows,  and  B  16  cows  ;  how  much 
should  each  pay  ? 

4.  Two  men  hired  a  pasture  for  $10  ;  A  put  in  8  cows 

3  months,  and  B  put  in  4  cows  4  months;  how  much 
should  each  pay  ? 

The  pasturage  of  8  cows  for  3  months  is  the  same  as  of 
24  cows  for  1  month,  and  the  pasturage  of  4  cows  for  4 
months  is  the  same  as  of  16  cows  for  1  month.  The  shares 
of  A  and  B,  therefore,  are  24  to  16,  as  in  the  former  ques- 
tion. Hence,  when  time  is  regarded  in  fellowship, — 
Multiply  each  one's  stock  by  the  time  he  continues  it  in  trade, 
and  use  the  product  for  his  share.  This  is  called  Double 
Fellowship.  Ans.  A  6  dollars,  and  B  4  dollars. 

5.  A  and  B  enter  into  partnership ;  A  puts  in  $100 
6  months,  and  then  puts  in   $50  more  ;  B  puts  in  $200 

4  months,  and  then  takes  out  $80  ;  at  the  close  of  the 
year  they  find  that  they  have  gained  $95 ;  what  is  the 
profit  of  each  ?  .        (  $43,711,  A's  share, 

^"^'  I  $51,288,  B's  share. 

6.  A,  with  a  capital  of  |500,  began  trade,  Jan.  1, 1826, 
and,  meeting  with  success,  took  in  B  as  a  partner,  with  a 
capital  of  600,  on  the  first  of  March  following  ;  four 
months  after,  they  admit  C  as  a  partner,  who  brought  $800 
stock  ;  at  the  close  of  the  year,  they  find  the  gain  to  be 
$700  ;  how  must  it  be  divided  among  the  partners  ? 

$250,  A's  share. 

Ans.  I  <5^250,  B's  share. 

C's  share. 


ALLIGATION. 

The  rule  of  Alligation  teaches  how  to  gain  the  mean 
tmlue  of  a  mixture  that  is  made  by  uniting  several  articles 
of  different  values. 


ALLIGATION.  229 

Alligation  Medial,  teaches  how  to  obtain  the  vahie,  (or 
mean  price,)  of  a  mixture,  when  the  quantities  and  prices 
of  the  several  articles  are  given. 

Rule. 
As  the  whole  mixture  is  to  the  whole  value,  so  is  any  part 
)f  the  composition,  to  its  mean  price. 

Examples. 
1.  A   farmer  mixed  15  bushels  of  r5fe,  at  64  cents  a 
bushel,  18  bushels  of  Indian  corn,  at  55  cts.  a  bushel,  and 
21  bushels  of  oats,  at  28  cts.  a  bushel ;  I  demand  what  a 
bushel  of  this  mixture  is  worth  ? 


hu.     cts.    $  cts. 

hu. 

$  cts.       hu. 

15  at  64=9,60 

As  54  : 

;  25,38  :   :   1 

18      55=9,90 

1 

21      28=5,88 

cts. 

—  54)25,38(.47  Ans. 

54  25,38 

2.  If  20  bushels  of  wheat  at  1  dol.  35  cts.  per  bushel, 
be  mixed  with  10  bushels  of  rye  at  90  cents  per  bushel, 
what  will  a  bushel  of  this  mixture  be  worth  ? 

Ans.  81,20  cts. 

3.  A  tobacconist  mixed  36  lb.  of  tobacco,  at  Is.  6d. 
per  lb.,  12  lb.  at  2s.  a  pound,  with  12  lb.  at  Is.  JOd.  per 
lb.  ;  what  is  the  price  of  a  pound  of  this  mixture  ? 

Ans.  Is.  8d, 

4.  A  grocer  mixed  2  C.  of  sugar  at  56s.  per  C.  and  1 
C.  at  43s.  per  C.  and  2  C.  at  50s.  per  C.  together ;  I  de- 
mand the  price  of  3  cwt.  of  this  mixture?       Ans.  £7. 13s. 

5.  A  wine  merchant  mixes  15  gallons  of  wine  at  4s. 
2d.  per  gallon,  with  24  gallons  at  6s.  8d.  and  20  gallons 
at  6s.  3d. ;  what  is  a  gallon  of  this  composition  worth  ? 

Ans.  5s.  lOd.  24|  qrs. 
Alligation  Alternate,  teaches  how  to  find  the  quantity  of 
each  article,  when  the  mean  price  of  the  whole  mixture, 
and  also  the  prices  of  each  separate  article  are  known. 

Rule. 
Reduce  the  mean  price  and  the  prices  of  each  separate 
article  to  the  same  order. 

20 


230  ARITHMETIC.      PART   THIRD. 

Connect  with  a  line  each  "price  that  is  less  than  the  mean 
price,  tcith  one  or  more  that  is  greater;  and  each  price 
greater  than  the  mean  price,  with  one  or  more  that  is  less. 

Write  the  difference  between  the  mean  price,  and  the  price 
cf  each  separate  article,  opposite  the  price  with  which  it  is 
connected ;  then  the  sum  of  the  differences,  standing  against 
any  price,  will  express  the  relative  quantity  to  he  taken  oj 
that  price. 

Examples. 

1.  A  merchant  has  several  kinds  of  lea  ;  some  at  8 
shillings,  some  at  9  shillings,  some  at  1 1  shillings,  and 
some  at  12  shillings  per  pound;  what  proportions  of  each 
must  he  mix,  that  he  may  sell  the  compound  at  10  s. 
per  pound  ? 

The  pupil  will  perceive,  that  there  may  be  as  many 
difterent  ways  of  mixing  the  simples,  and,  consequently 
as  many  different  answers,  as  there  are  different  ways  of 
linking  the  several  prices. 

Operations. 


10= 


Ihs 

8s. ,-2 

9s.—, 

-1 

lis > 

-1 

12s.         J 

-2 

Or,  ^    8 f — 2+1=3 

1       =1 
II. 
12- 


r  10  ^  9- 


-1+2=3 
-2       =2 


Here  the  prices  of  the  simples,  are  set  one  directly 
under  another,  in  order,  from  least  to  greatest,  and  the 
mean  rate,  (10s.)  written  at  the  left  hand.  In  the  first  way 
of  linking,  we  take  in  the  proportion  of  2  pounds  of  the 
teas  at  8  and  12s.  to  1  pound  at  9  and  lis.  In  the  second 
way,  we  find  for  the  answer,  3  pounds  at  8  and  lis.  to  1 
pound  at  9  and  12s. 

2.  What  proportions  of  sugar,  at  8  cents,  10  cents,  and 
14  cents  per  pound,  will  compose  a  mixture  worth  12 
cents  per  pound  ? 

Ans.  In  the  proportion  of  2  lbs.  at  8  and  10  cents,  to  6 
lbs.  at  14  cents. 

Note.  As  these  quantities  only  express  the  proportions 
of  each  kind,  it  is  plain,  that  a  compound  of  the  same  mean 
price  will  be  formed  by  taking  3  times,  4  times,  one  half, 
or  any  proportion,  of  each  quantity.     Hence, 


ALLIGATION.  231 

When  the  quantity  of  one  simple  is  given,  after  finding 
tlie  proportional  quantities,  by  the  above  rule,  we  may  say, 
As  the  PROPoiiTioNAL  quantity  :  is  to  the  given  quantity  :  : 
.so  is  each  of  the  other  ruopoRxioNAL  quantities  :  to  the  re- 
QUIRED  quantities  of  each. 

3.  If  a  man  wishes  to  mix  1  gallon  of  brandy  worth 
16s.  with  rum  at  9s.  per  gallon,  so  that  the  mixture  may 
be  worth  lis.  per  gallon,  how  much  rum  must  he  use  ? 

Taking  the  differences  as  above,  we  find  the  propor- 
>ions  to  be  2  of  brandy  to  5  of  rum  ;  consequently,  1  gai- 
lon'of  brandy  will  require  21  gallons  of  rum. 

Ans.  2^  gallons. 

4.  A  grocer  has  sugars  worth  7  cents,  9  cents,  and  12 
cents  per  pound,  which  he  would  mix  so  as  to  form  a  com- 
pound, worth  10  cents  per  pound  ;  what  must  be  the  pro- 
portions  of  each  kind  ? 

Ans.  2  lbs.  of  the  first  and  second,  to  4  lbs.  of  the  3d  kind. 

5.  If  he  use  1  lb.  of  the  first  kind,  how  much  must  he 

take  of  the  others? if  4  lbs.,  what  ? if  6  lbs., 

what  ? if  10  lbs.,  what  ? if  20  lbs.,  what  1 

Ans.  to  tho  last,  20  lbs.  of  the  2d,  and  40  of  the  3d. 

6.  A  merchant  has  spices  at  16d.  20d.  and  32d.  per 
pound  ;  he  would  mix  5  pounds  of  the  first  sort  with  the 
others,  so  as  to  form  a  compound  worth  24d.  per  pound  ; 
how  much  of  each  sort  must  he  use  ? 

Ans.  5  lbs.  of  the  second,  and  7i  lbs.  of  tho  third. 

7.  How  many  gallons  of  water,  of  no  value,  must  be 
mixed  with  60  gallons  of  rum,  worth  80  cents  per  gallon, 
1.0  reduce  its  value  to  70  cents  per  gallon  ?     Ans.  8i  galls. 

8.  A  man  would  mix  4  bushels  of  wheat,  at  81,50  per 
bushel,  rye  at  81,1G,  corn  at  8,75,  and  barley  at  8,50,  so 
as  to  sell  the  mixture  at  8,84  per  bushel  ;  how  much  of 
each  may  he  use  1 

When  the  quantity  of  the  compound  is  given,  we  may 
say.  As  the  sum  of  the  proportional  quantities,  found  by 
the  ahoverule,  is  to  the  quantity  required,  so  is  each  pro- 
portional quantity,  found  hy  the  rule,  to  the  required 
quantity  of  each. 

9.  A  man  would  mix  100  pounds  of  sugar,  some  at  8 
cents,  some  at  10  cents,  and  some  at  14  cents  per  pound, 


232  ARITHMETIC,       PART  THIRD. 

SO  that  the  compound  may  be  worth  12  cents  per  pound  ; 
how  much  of  each  kind  must  he  use  ? 

We  find  the  proportions  to  be,  2, 2,  and  6.  Then,  2+2 
+6=  10,  and  ^  2  :  20  lbs.  at    8  cts.  ) 

10  :  100  :  :  ^  2  :  20  lbs.  at  10  cts.  \  Ans. 
(6  :  60  lbs.  at  14  cts.  ^ 

10.  How  many  gallons  of  water,  of  no  value,  must  be 
mixed  with  brandy  at  ^1,20  per  gallon,  so  as  to  fill  a  ves- 
sel  of  75  gallons,  which  may  be  worth  92  cents  per  gal.  ? 

Ans.   17i  gallons  of  water  to  57i  gallons  of  brandy. 

11.  A  grocer  has  currants  at  4d.,  6d.,  9d.,  and  lid.  per 
lb.  ;  and  he  would  make  a  mixture  of  240  lbs.,  so  that 
the  mixture  may  be  sold  at  8d.  per  lb. ;  how  many  pounds 
of  each  sort  may  he  take  ? 

Ans.  72,  24,  48,  and  96  lbs.,  or  48,  48,  72,  72,  &c. 
Note.  This  question  may  have  five  different  answers. 


DUODECIMALS. 

Duodecimal  is  derived  from  the  Latin  word  duodecim, 
signifying  twelve. 

They  are  fractions  of  a  foot,  which  is  supposed  to  be 
divided  into  twelve  equal  parts  called  immes,  marked  thus, 
(').  Each  prime  is  supposed  to  be  subdivided  into  12  equal 
parts  called  seconds,  marked  thus,  (").  Each  second  is 
also  supposed  to  be  divided  into  twelve  equal  parts  called 
thirds,  marked  thus  ('"),  and  so  on  to  any  extent. 

It  thus  appears  that 

1'  an  inch  or  prime  is  Jg  of  a  foot. 

1"  a  second  is  yL  of  J^  or  j\^  of  a  foot. 

r"  a  third  is  J^  of  J^  of  y^,  or  -j-J^g  of  a  foot,  &;c. 

Whenever  therefore  any  number  of  seconds  (as  5")  are 
mentioned,  it  is  to  be  understood  as  so  many  j}j  of  a  foot, 
and  so  of  the  thirds,  fourths,  &c. 

Duodecimals  are  added  and  subtracted  like  other  com- 
pound numbers,  12  of  a  less  order  making  1  of  the  next 
higher,  thus, 

12""  fourths  make  I  third  1"'. 

12"' thirds  make  1  second  1". 

12"  seconds  make  1  prime  or  inch  1'. 


DUODECIMALS.  233 

12'  inches  or  primes,  make  1  foot. 
The  addition  and  subtraction  of  Duodecimals  is  the 
same  as  other  compound  numbers. 

These  marks  '  "  '"  ""  are  called  indices. 


Multiplication  of  Duodecimals. 

Duodecimals  are  chiefly  used  in  measuring  surfaces  and 
solids. 

How  many  square  feet  in  a  board  16  feet  7  inches  long, 
and  1  foot  3  inches  wide  ? 

Note.  The  square  contents  of  any  thing  are  found  by 
multiplying  the  length  into  the  breadth. 

The  following  example  is  explained  above. 


Examples. 

16 

7' 

1 

3' 

IG 

7 

4 

r      9 

20         8'         9" 

It  is  generally  more  convenient  to  multiply  by  the  higher 
orders  of  the  multiplier /rsi. 

Thus  we  begin  and  multiply  the  multiplicand  first  by  the 
1  foot,  and  set  down  the  answers  as  above. 

We  then  multiply  by  the  3'  or  j\  of  a  foot.  16  is  chan- 
ged  to  a  fraction,  thus  ^-^,  and  this  multiplied  by  /^  is  a|, 
or  48',  which  is  4  feet,  (for  there  are  12'  in  every  foot,) 
and  is  set  under  that  order. 

We  now  multiply  7'  (or  /j )  by  3'  (or  y^-)  and  the  answer 
is /J- or  21". 

This  is  r  to  set  under  the  order  of  seconds,  and  9"  (yfy) 
to  be  set  under  the  order  of  thirds. 

The  two  products  are  then  added  together,  and  the 
answer  is  obtained,  which  is  20  feet  8  primes  9  seconds. 

Another  example  will  be  given  in  which  the  cubic  con- 
20* 


284  ARITHMETIC.      PART  THIRD. 

tents  of  a  block  are  found  by  multiplying  the  length,  breadth 
and  thickness  together. 

How  many  solid  feet  in  a  block  15  ft.  8'  long,  1  ft.  5' 
wide,  and  1  ft.  4'  thick  7 


Length, 
Breadth, 

1 

15 

1 

Operation, 

8' 
5' 

4" 

15 

6 

8' 
6' 

Thickness 

22 

1 

2' 
4' 

4" 

22 

7 

2' 
4' 

4" 
9" 

4'" 

Ans. 

29 

r 

1" 

4'" 

Let  this  example  be  studied  and  understood  before  the 
rule  is  learned.  If  any  difficulty  is  found,  let  both  multi- 
plier and  multiplicand  be  expressed  as  Vulgar  Fractions, 
and  then  multiply. 

In  duodecimals  it  is  always  the  case  that  the  product  of 
two  orders,  will  belong  to  that  order  which  is  made  by  ad- 
ding the  indices  of  the  factors. 

Rule. 

W7nte  the  figures  as  in  the  addition  of  compound  numbers. 
Multiply  by  the  higher  orders  of  the  jnultiplier  first,  remem- 
bering that  the  product  of  two  orders  belongs  to  the  order  de- 
noted by  the  sum  of  their  indices. 

If  any  product  is  large  enough  to  contain  units  of  a  higlier 
order,  change  them  to  a  higher  order,  and  place  them  where 
they  belong. 

ExAltlFLES. 

How  many  square  feet  in  a  pile  of  boards  12  ft.  8'  long, 
and  13'  wide  ? 


INVOLUTION.  235 


What  is  the  product  of  371  ft.  2'  6"  multiplied  by  Ibl 
ft.  1'  9"  ?  Ans.  G7242  ft.  10'  1  "  4'"  6"". 

If  a  floor  be  10  ft.  4'  5"  long,  and  7  ft.  8'  G"  wide,  what 
is  its  surface  ?  Ans.  79  ft.  11'  0"  6'"  6"". 

What  is  the  solidity  of  a  wall  53  ft.  6'  long,  10  ft.  3' 
high,  and  2  ft.  thick  ?  Ans.  1096Kt. 


INVOLUTION. 

When  a  number  is  multiplied  into  itself,  it  is  said  to  be 
involved,  and  the  process  is  called  Involution. 

Thus,  2X2x2  is  8.  Here  the  number  2  is  multiplied 
into  itself  twice. 

The  'product  which  is  obtained  by  multiplying  a  number 
into  itself,  is  called  a  Power. 

Thus,  when  2  is  multiplied  into  itself  once,  it  is  4,  and 
this  is  called  the  second  power  of  2.  If  it  is  multiplied  into 
\ise\i  twice  (2x2x2=8)  the  answer  is  8,  and  this  is  called 
the  third  power. 

The  number  which  is  involved,  is  called  the  Root,  or 
first  power. 

Thus,  2  is  the  root  of  its  second  power  4,  and  the  root  of 
its  third  power  8. 

A  power  is  named,  or  numbered,  according  to  the  number 
of  times  its  root  is  used  as  a  jactor.  Thus  the  number  4 
is  called  the  second  power  of  its  root  2,  because  the  root 
is  twice  used  as  a  factor  ;  thus,  2x2=4. 

The  number  8  is  called  the  third  power  of  its  root  2  ; 
because  the  root  is  used  three  times  as  a  factor ;  thus, 
2X2x2=8. 

The  method  of  expressing  a  power,  is  by  \vriting  its 
root,  and  then  above  it  placing  a  small  figure,  to  show  the 
number  of  times  that  the  root  is  used  as  a  factor. 

Thus  the  second  power  of  2  is  4,  but  instead  of  writing 
the  product  4,  we  write  it  thus,  2^ . 

The  third  power  of  2  is  written  thus,  2^. 

The  fourth  power  of  2  is  IG,  and  is  written  thus,  2*. 

The  small  figure  that  indicates  the  number  of  times  that 
the  root  is  used  as  a  factor,  is  called  the  Index,  or  Expo- 
nent. 


236  ARITHMETIC.      PART   THIRD. 

The  different  powers  have  other  names  beside  their 
numbers. 

Thus,  the  second  power  is  called  the  Square. 

The  third  power  is  called  the  Cube. 

The  fourth  power  is  called  the  Biquadrate. 

The  fifth  power  is  called  the  Sursolid. 

The  sixth  power  is  called  the  Square.cuhed. 

Powers  are  indicated  by  exponents.  When  a  power  is 
actually  found  by  multiplication,  involution  is  said  to  be 
performed,  and  the  number  or  root  is  involved. 

Rule  of  Involution. 

To  involve  a  number,  multiply  it  into  itself,  as  often  as 
there  are  units  in  the  exponent,  save  once. 

Note. — The  reason  why  it  is  multiplied  once  less  than 
there  are  units  in  the  exponent,  is,  that  the  first  time  the 
number  is  multiplied,  the  root  is  used  twice  as  a  factor  ; 
and  the  exponent  shows,  not  how  many  times  we  are  to 
multiply,  but  how  many  times  the  root  is  used  as  a  factor. 

1.  What  is  the  cube  of  5  ?  Ans.  5x5  X5=125 

2.  What  is  the  4th  power  of  4  ?  Ans.  256 

3.  What  is  the  square  of  14?  Ans.  196 

4.  WHiat  is  the  cube  of  6  ?  Ans.  216 

5.  What  is  the  5th  power  of  2  ?  Ans.  32 

6.  What  is  the  7th  power  of  2  ?  Ans.  128 

7.  W^hat  is  the  square  of  i  ?  Ans.  i 

8.  What  is  the  cube  of  f  ?  Ans.  ■^\ 
A  Fraction  es  involved,  by  involving  both  numerator  and 

denominator. 

9.  What  is  the  fourth  power  off  ?  Ans.  g%'_ 

10.  What  is  the  square  of  51  ?  Ans.  30i 

11.  What  is  the  square  of  301  ?  Ans.  915-pV 

12.  Perform  the  involution  of  85.  Ans.  32,768 

13.  Involve   3%,  j^,  and  f  to  the  third  power  each. 

Xna  64         .1331.5  12 

■^"'•5U3T9'     T7  28'    T29 

14.  Involve  21  P.  Ans.  9,393,931 

15.  Raise  25  to  tlie  fourth  power.  Ans.  390,625 

16.  Find  the  sixth  power  of  1.2.  Ans.  2.985,984 


EVOLUTION.  237 

EVOLUTION. 

Evolution  is  the  process  of  finding  the  root  of  any  num- 
ber;  that  is,  of  finding  that  number  which  multiplied  into 
itself,  will  produce  the  given  nuhiber. 

The  Square  Root,  or  Second  Root,  is  a  number  which  be- 
ing squared  (i.  e.  multiplied  once  into  itself)  will  produce 
the  given  number.  It  is  expressed  either  by  this  sign,  put 
before  a  number,  thus  \/4,  or  by  the  fraction  I  placed 

i_ 
above  a  number  thus,  4-. 

The  Cube  Root,  or  Third  Root,  is  a  number,  which  be- 
ing  cubed,  or  multiplied  by  itself  twice,  will  produce  the 

3  1 

given  number.     It  is  expressed  thus,  \/12  ;  or  thus,  123. 
All  the  other  roots  are  expressed  in  the  same  manner. 

4 

Thus  the  fourth  root  has  this  sign  V  put  before  a  number, 
or  else  i  placed  above  it, 

6 

The  sixth  root  has  -/  before  it,  or  J-  above  it,  &c. 

There  are  some  numbers  whose  roots  cannot  be  pre- 
cisely obtained ;  but  by  means  of  decimals,  we  can  cp- 
proximate  to  the  number  which  is  the  root. 

Numbers  whose  roots  can  be  exactly  obtained,  are 
called  rational  numbers. 

Numbers  whose  precise  roots  cannot  be  obtained,  are 
called  surd  numbers. 

When  the  root  of  several  numbers  united  by  the  sign 
-f-  or  —  is  indicated,  a  mnculum,  or  line  is  drawn  from 
the  sign  of  the  root  over  the  numbers.  Thus,  the  square 
root  of  36 — 8  is  written  ^3(i— 8. 

The  root  of  a  rational  number,  is  a  rational  root,  and  the 
root  of  a  surd  number,  is  a  surd  root.  ' 

It  is  very  necessary  for  practical  purposes,  to  be  able  to 
find  the  amount  oi  surf  ace  there  is  in  any  given  quantity. 
For  instance,  if  a  man  has  250  yards  of  matting,  which  is 
•2  yards  wide,  how  much  surface  will  it  cover  ? 

The  rule  for  finding  the  amount  of  surface,  is  to  multiply 
the  length  by  the  breadth,  and  this  will  give  the  amount  of 
square  inches,  feet,  oy  yards. 

It  is  important  for  the  pupil  to  learn  the  distinction  be- 
tween a  square  quantity,  and  a  certain  extent  that  is  in  the 


238 


ARITHMETIC.        PART   THIRD. 


¥■ 

I 

form  of  a  square.     For  example,  Jour  square  inches,  and 

Jour  inches  square  are  different  quantities. 

A  Four  square  inches  may  be   represented 

in  Fig.  A.  In  this  figure  there  are  four 
square  inches,  but  it  makes  a  square  which  is 
only  two  inches  on  each  side,  or  a  two  inch 
square. 

"•  A  Jour  inch  square  may  be  re- 

presented by  Fig.  B. 

Here  the  sides  of  the  square  are 
four  inches  long,  and  it  is  called 
a  Jour  inch  square.  But  it  con- 
tains sixteen  square  inches.  For 
when  the  four  inch  square  is  cut 
into  pieces  of  each  an  inch  square 
it  will  make  sixteen  of  them. 
A  Jour  inch  square  then,  is  a  square  whose  sides  are  four 
inches  long. 

Four  square  inches  are  four  squares  that  are  each  an 
inch  on  every  side.  . 

When  we  wish  to  find  the  square  contents  of  any  quan- 
tity, we  seek  to  know  how  many  square  inches,  or  feet,  or 
yards,  there  are  in  the  quantity  given,  and  this  is  always 
found  by  multiplying  the  length  by  the  breadth. 

When  the  length  and  breadth  of  any  quantity  are  given, 
we  find  its  square  contents,  or  the  amount  of  surface  it  will 
cover,  by  multiplying  the  length  by  the  breadth. 

What  are  the  square  contents  of  223  yds  of  carpeting  | 
Avide  ? 

What  are  the  sq.  contents  of  249  yds  of  matting  |  wide  ? 
If  any  quantity  is  placed  in  a  square  form,  the  length  of 
one  side  is  the  square  root  of  the  square  contents  of  this 
figure.  Thus  in  the  preceding  example,  B,  the  square 
contents  of  the  figure  are  16  square  inches.  The  side  of 
the  square  is  4  inches  long  ;  and  4  is  the  square  root  of  16. 
The  square  root,  therefore,  is  the  length  of  the  sides  of  a 
square,  made  by  the  given  quantity. 

If  we  have  one  side  of  a  square  given,  by  the  process  of 
Involution,  we  find  what  are  the  square  contents  of  the  quan. 
tity  given. 


EXTRACTION  OF  THE  SQUARE  ROOT.       239 

If  on  the  contrary,  we  have  the  square  contents  given, 
by  the  process  of  Evolution,  we  find  what  is  the  length  of 
one  side  of  the  square,  which  can  be  made  by  the  quantity 
given. 

Thus  if  we  have  a  square  whose  side  is  four  inches,  by 
Involution  we  find  the  surface,  or  square  contents  to  be  16 
square  inches. 

But  if  we  have  16  square  inches  given,  by  Evolution  we 
find  what  is  the  length  of  one  side  of  the  square  made  by 
these  16  inches. 


EXTRACTION  OF  THE  SQUARE  ROOT. 

Extracting  the  square  root  is  finding  a  number,  which, 
muUiphed  into  itself,  will  produce  the  given  number ;  or, 
it  is  finding  the  length  of  one  side  of  a  certain  quantity, 
when  that  quantity  is  placed  in  an  exact  square. 

It  will  be  found  by  trial,  that  the  root  always  contains 
just  half  as  many,  or  one  figure  more  than  half  as  many 
figures  as  are  in  the  given  quantity.  To  ascertain,  there- 
fore,  the  number  of  figures  in  the  required  root,  we  point 
off"  the  given  number  into  periods  of  two  figures  each,  be- 
ginning at  the  right,  and  there  will  always  be  as  many 
figures  in  the  root  as  there  are  periods. 

1.  What  is  one  side  of  a  square,  containing  784  square 
feet? 

784(2     Pointing  off" as  above,  we  find  that  the  root  will 
4  consist  of  two  figures,  a  ten  and  a  unit. 

We  now  take  the  highest  peri- 

384  od  7  (hundreds),  and  ascertain 

how  many  feet  there  will  be  in 
Fig.  1.  the  largest   square  that  can  be 

made  of  this  quantity,  the  sides 
of  which  must  be  of  the  order  of 
tens.  No  square  larger  than  4 
(hundreds)  can  be  contained  in  7 
(hundreds),  the  sides  of  which 
will  be  each  20  feet  (because  20x 
20=400).  These  20  feet  (or  2 
tens)  being  sides  of  the  square 


B 

20 
20 

400 


20  feet.  2ire  placed  in  the  quotient  as  the 

first  figure  of  the  root. 


240 


AKITHMETIC.       PART    THIRD. 


This  square  may  be  represented  by  Fig  1. 

We  now  take  out  the  400  from  700,  and  300  square  feet 
remain.  These  are  added  to  the  next  period  (84  feet), 
making  384,  which  are  to  be  arranged  around  the  square 
B,  in  such  a  way  as  not  to  destroy  its  square  form ;  conse- 
quently  the  additions  must  be  made  on  two  sides. 

To  ascertain  the  breadth  of  these  additions,  the  384 
must  be  divided  by  the  length  of  the  two  sides  (20-|-20), 
and  as  the  root  already  found  is  one  side,  we  double  this 
root  for  a  divisor,  making  4  tens  or  40,  for  as  40  feet  is 
the  length  of  these  sides,  there  will  be  as  many  feet  in 
breadth,  as  there  are  forties  in  384.  The  quotient  arising 
from  the  division  is  8,  which  is  the  breadth  of  the  addition 
to  be  made,  and  which  is  placed  in  the  quotient,  after  the 
4  tens.  .    . 

784(28  Root. 
4 


48 


384 
384 

000 


Fig.  2. 


20  feet. 

8  feet. 

C 

E 

r« 

00 

20X8=1G0 

8x8=64 

*i> 

B 

o 

-t-" 

X 

^ 

II       ^ 

o 

o 

20X20=400 

20  feet. 


8  feet. 


But  it  will  be  seen  by  Fig.  2,  that  to  complete  the  square, 
the  corner  E  must  be  filled  by  a  small  square,  the  sides  of 
which  are  each  equal  to  the  widtJi  of  C  and  D,  that  is,  8 


EXTRACTION  OF  THE  SQUARE  ROOT.        241 

feet.  Adding  this  to  the  4  tens,  or  40,  we  find  that  the 
whole  length  of  the  addition  to  be  made  around  the  square 
B,  is  48  feet,  instead  of  40.  This  multiplied  by  its  breadth, 
8  feet  (the  quotient  figure),  gives  the  contents  of  the  whole 
addition,  viz.  384  feet. 

As  there  is  no  remainder,  the  work  is  done,  and  28  feet 
is  the  side  of  the  given  square. 

The /»ron/"  may  be  seen  by  involution,  thus  ;  28X28= 
784  ;  or  it  may  be  proved,  by  adding  together  the  several 
parts  of  the  figure,  thus  ; 

B  contains  400  feet. 
C         "       160  " 
D         "       160  " 
E         "         64  « 

Proof    784 

If,  in  any  case,  there  is  a  remainder,  after  the  last  period 
is  brought  down,  it  may  be  reduced  to  u  decimal  fraction, 
by  annexing  two  cipiiers  for  a  new  period,  and  the  same 
process  continued. 

Whenever  any  dividend  is  too  smallto  contain  the  divi- 
sor,  a  cipher  must  be  placed  in  the  root,  and  another 
period  brought  down. 

From  the  above  illustrations,  we  see  the  reasons  for  the 
following  rule. 

Rule  for  Extracting  the  Square  Root. 

1 .  Point  of  the  given  number,  into  periods  of  iico figures 
each,  beginning  at  the  right. 

2.  Find  the  greatest  square  in  the  first  left  hand  period, 
and  subtract  it  from  that  period.  Place  the  root  oj  this 
square  in  the  quotient.  To  the  remainder  bring  doion  the 
next  period  for  a  dividend. 

3.  Double  the  root  already  found  {understanding  a  cipher 
at  the  right)  for  a  divisor.  Divide  the  dividend  by  it,  and 
place  the  quotient  figure  in  the  root,  and  also  in  the  divisor. 

4.  Multiply  the  divisor,  thus  increased,  by  the  last  figure 
of  the  root,  and  subtract  the  product  from  the  dividend.  To 
the  remainder  bring  down  the  next  period,  for  a  new  divi- 

21 


242 


ARITHMETIC.       PART  THIRD. 


dend.     Double  the  root  already  found,  for  a  new  divisor, 
and  proceed  as  before. 

Examples. 


What  is  the  square  root  of  998001  ? 
998001(999  Rooi. 

81 


189)1880 
1701 


1989)17901 
17901 


000 


Find  the  sq.  root  of  784.  A.  28.  Of  07(J.  A.  2U. 
Of  625.  A.  25.  Of  487,204.  A.  698.  Of  638,401. 
A.  779.  Of  556,510.  A.  746.  Of  441.  A.  21.  Ot 
1024.  A.  32.  Of  1444.  A.  38.  Of  2916.  A. 
Of  6241.     A.  79.     Of  9801.     A.  99.     Ot   17,956. 


134.     Of  32,761.     A.  181.     Of  39,601.     A.   199. 
488,601.     A.  699. 

Find  the  sq.  root  of  69.  A.  8.3066239.  Of  83. 


54. 
A. 

Of 


9.1104336. 

17.2916165. 

16.7928556. 

18.7349940. 

31.2889757. 

31.6069613, 

26,2106848. 


Of  97, 
Of  222. 
Of  394. 
Of  699. 
Of  989. 
Of  397. 
Of  892. 


A. 
A. 
A. 
A. 
A. 
A. 


9.8488578. 
14.8996644. 
19.6494332. 
26.4386081. 
31.4483704. 
19.9248588. 


Of   299; 

Of  282. 
Of  351. 
Of  979. 
Of  999. 

Of  687. 


A. 
A. 
A. 
A. 
A. 
A. 
A. 


A.  29.866,3600. 

It  was  shown  in  the  article  on  Involution,  that  a  fraction 
is  involved  by  involving  both  numerator  and  denominator, 
hence  to  find  the  root  of  a  fraction,  extract  the  root  loth  of 
numerator  and  denominator.  If  this  cannot  be  done,  the 
fraction  may  be  reduced  to  a  decimal,  and  its  root  ex- 
tracted. 

What  is  the  square  root  of  f|  ?      A 

k      4  ox        Of  3-311  «S-  9       4.    AS  7,       Of 


oinmv-  A. 


6  93" 
Of  6.0  6.8;  4  1   ? 


^*    2  4  9  0  0  1     • 

4.303.3.4   7 
'463025     • 

A.    11^. 


A.    s^-'. 


'JTT' 


EXTRACTION  OF  THE  TUBE  ROOT.  243 


A.    .8660254. 

Of 

s 

1  2* 

A. 

A 

4.168333. 

Of 

_3_ 
8  0 

A. 

A.     .83205. 

Of 

S 
6  U  • 

A. 

Find  the  sq.  root  of 
.645497.       Of    17f. 
.193649167.        Of   /^ 
.288617394+ 


EXTRACTION  OF  THE  CUBE  ROOT. 

A  Cube  is  a  solid  body,  having  six  equal  sides,  each  of 
which  is  an  exact  sciuare.  Thus  a  sohd,  which  is  1  foot 
long,  1  foot  high,  and  1  foot  wide,  is  a  cubic  joot ;  and  a 
solid  whose  length,  breadth,  and  thickness  are  each  I  yard, 
is  called  a  cubic  yard. 

The  root  of  a  cube  is  always  the  length  of  one  of  its 
sides  ;  for  as  the  length,  breadth,  and  thickness  of  such  a 
body  are  the  same,  the  length  of  one  side,  raised  to  the 
third  power,  will  show  the  contents  of  the  whole. 

Extracling  the  Cube  Root  of  any  quantity,  therefore,  is 
finding  a  number,  which  multiplierl  into  itself,  twice,  will 
produce  that  quantity ; — or  it  is  finding  the  length  of  one 
side  of  a  given  quantity,  when  that  quantity  is  placed  in 
an  exact  cube. 

To  ascertain  the  number  of  figures  in  a  cube  root,  wc 
point  otr  the  given  number,  into  periods  of  three  figures 
each,  beginning  at  the  right,  and  there  v/ill  be  as  many 
figures  in  the  required  root  as  there  are  periods. 

1.  What  is  the  length  of  one  side  of  a  cube,  containing 
32768  solid  feet?  .      . 

32768(3 
27 


5768 


Pointing  off  as  above,  we  find  there  will  be  two  figures 
in  the  root,  a  ten  and  a  unit. 


244 


ARITHiMKTIC.       PART    THIRD, 


We   now  take  the  highest 
period,  32  (thousands),  and 
ascertain  what  is  the  largest 
cube   that  can  be  contained 
in  this  quantity,  the  sides  of 
which  will  be  ot''the  order  of 
tens.    No  cube  larger  than  2? 
(thousands)  can  be  contained 
in  32  (thousands).   The  sides 
of  this  are  3  tens  or  30  (be- 
cause    30x30x30=27,000) 
which  are  i>laced  as  the  first  figure  of  the  root. 
This  cube  may  be  represented  bv  Fig  1. 
We  now  take  the  27000  from  32000,  and  5000  solid  feet 
remain.     These  are  added  to  the  next  period  (768),  ma- 
king 5768,  which  are   to  be   arranged  around  the   cubic 
figure  1,  in  such  a  way  as  not  to  destro}'  its  cubic  form  ; 
consequently  the  addition  must  be  made  to  three  of  its 
sides. 

We  must  now  ascertain,  what  will  be  the  thickness  of 
the  addition  made  to  each  side.  This  will  of  course  de- 
pend upon  the  surface  to  be  covered.  Now  the  length  of 
one  side  has  been  shown  to  be  30  feet,  and,  as  in  a  cube, 
the  length  and  breadth  of  the  sides  are  equal,  multiplying 
the  length  of  one  side  into  itself  will  show  the  surface  of 
one  side,  and  this  multiplied  by  3,  the  number  of  sides,  gives 
the  contents  of  the  surface  of  the  three  sides.  Thus  30x 
30=900,  which  multiplied  by  3=2700  feet. 

Now  as  we  have  3768  solid  feet  to  be  distributed  upon  a 
surface  of  2700  feet,  there  will  be  as  many  feet  in  the 
thickness  of  the  addition,  as  there  are  twenty-seven  hun- 
dreds in  3768.  2700  is  contained  in  3768  twice  ;  there- 
fore 2  feet  is  the  thickness  of  the  addition  made  to  each  of 
the  three  sides. 

By  multiplying  this  thickness,  by  the  extent  of  surface 
(2700x2)  v,e  find  that  there  are  5400  solid  feet  contained 
in  these  additions. 


EXTRACTION  OF  THE  CUBE  ROOT. 


245 


32768(32 
27 

2700)5708 

5400 
380 

8 

5768 


0000 


But  if"  we  examine  Fig.  2  we  shall 
find  that  these  additions  do  not  com- 
plele  the  cube,  for  the  three  cornfirs 
a  a  a  need  to  be  iilled  by  blocks  of  the 
same  length  as  the  sides  (30  feet)  and 
of  the  same  breadth  and  thickness  as 
the  previous  additions  (viz.  2  teet). 

Now  to    find  tiic   solid   contents    of 
these  blocks,    or  the  number  of  feet 
required  to  fill  these  corners,  wo  multi- 
ply the  length,  breadth,  and  thickness 
of  one  block  together,  and  then  nivdtiply 
this  product  by  3,  the  num- 
ber of  blocks.      Thus,  the 
breadth    and    thickness   of 
each  block  has  been  "shown 
to  be  'Z  feet  ;  2  X  2=r4,  and 
this  multiplied  by  30  (the 
}:30  kngth)=l20,  which  is  the 
solid  contents  of  o/ie  block. 
But  in  ikree,  there  will  be 
three  times  as  many  solid 
feet,  or  300.  which    is  the 
number  recpiired  to  fill  the 
deficiences. 

In  other  words,  we  square 
the  last  quotient  figure  (2) 
multiply  the  product  by  the 
first  figure  of  the  quotient 
(3  tens)  and  then  nuiltiply 
the  last  product  by  3,  the 
number  of  deficiencies. 

But  bv  examining  Fig.  3, 
it  appears  that  the  figure  is 
not  yet  complete,  but  that  a 
small  cube  is  still  wanting, 
where  the  blocks  last  added 
meet.  The  sides  of  this  small  cube,  it  will  be  seen,  are 
each  equal  to  the  width  of  these  blocks,  that  is,  2  feet. 
If  each  side  is  2  feet  long,  the  whole  cube  must  contain  8 

21* 


246 


ARITHMETIC.       PART  THIRD. 


Fig.  4. 
32  fret. 


solid  feet  (because  2x2x2 
=8),  and  it  will  be  seen 
by  Fig.  4,  that  this  just  fills 
the  vacant  corner,  and 
completes  the  cube. 

We  have  thus  found, 
that  the  additions  to  be 
made  around  the  large 
cube  (Fig.  I)  are  as  fol- 
ovvs. 


82  teet. 
5400  solid  feet  upon  three  sides,  (Fig.  2). 
3G0     "       "     to  fill  the  corners  a  a  a. 

8     "       "     to  fill  the  deficiency  in  Fig,  3. 
Now  if  these  be  added  together,  their  sum  will  be  576S 
solid  feet,   which  subtracted  from  the  dividend  leave  no 
remainder  and  the  work  is  done.     32  feet  is  therefore  the 
length  of  one  side  of  the  given  cube. 

The  proof  may  be  seen  by  involving  the  side  now  found 
to  the  third  power,  thus  ;  32x32x32=  32768  ;  or  it  may 
be  proved  by  adding  together  the  contents  of  the  several 
parts,  thus, 

27000  feet=contents  of  Fis.  1. 
5400  "     ^addition  to  three  sides. 
•    360  "    ^addition  to  fill  the  corners  a  a  a. 
8  "    =additionto  fill  the  corner  in  Fig.  3. 

32768     Proof. 
From  these  illustrations  we  see  the#easons  for  the  fol- 
loviing  rule. 

Rule  for  extracting  the  Cube  Root. 

1 .  Point  off  the  given  number,  into  periods  of  three  figures 
each,  beginning  at  the  right. 

2.  Fi7id  the  greatest  cube  in  the  left  hand,  period,  and 
svhiract  it  from  that  period.  '  Place  the  root  in  the  quotient, 
and  to  the  remainder  bring  down  the  next  period,  for  a  divi- 
dend. 

3.  Square  the  root  already  found  [understanding  a  cipher 
at  the  right)  and  multiply  it  by  3  for  a  divisor. 


\ 


EXTRACTION  OF  THE  CUBE  ROOT.         247 

Divide  the  dividend  hy  the  divisor,  and  place  the  quotient 
for  the  next  figure  of  the  root. 

4.  Multiply  the  divisor  by  this  quotient  figure.  Midtiply 
this  quotient  fgure  hy  iJie  former  figure  or  figures  of  the  root. 
Finally  cube  this  quotient  figure,  and  add,  these  three  results 
together  for  a  subtrahend. 

5.  Subtract  the  subtrahend  from  the  dividend.  To  the 
remainder  bring  down  the  next  period,  jor  a  new  dividend, 
and  proceed  as  before. 

If  it  happens  in  any  case,  that  the  divisor  is  not  con- 
tained in  the  dividend,  or  if  there  is  a  remainder  after  the 
last  period  is  brouoht  down,  the  same  directions  may  be 
observed,  that  were  given  respecting  the  square  root. 
(See  page  241.) 

Examples. 

What  is  the  cube  root  of  373248  ? 


373248(72 

343 

702x3=14700)30248(First  Dividend. 

29400 

22  X70X3-      840 

23=         8 

30248     Subtrahend. 

0000 

Find  the  cube  root  of  941,t92,000. 

A.  980. 

Of 

958,565,256.     A.  986.     Of 478,21 1,768. 

A.  782. 

Of 

494,913,671.     A.  791.     Of  445,943,744. 

A.  764. 

Of 

196,122.941.     A.  581.     Of  204,336,469. 

A.  589. 

Of 

57,512,450.     A.  386.      Of  6,751,269. 

A.     189. 

Of 

39,651,821.      A.  341.       Of  42,508,549. 

A.   349. 

Of 

510,082,399.     A.  799.     Of  469,097,433. 

A.  777. 

Find  the  cube  root  of  7.     A.   1.912933. 

Of  41. 

A. 

3.448217.  Of  49.  A.  3.6.59306.  Of  94.  A.  4.546836. 
Of  97.  A.  4.610436.  Of  199.  A.  5.838272.  Of  179. 
A.    5.635741.     Of  389.      A.  7.299893.      Of  364.     A. 


248  ARITHMETIC.       PART  THIRD, 

7.140037.  Of  499.  A.  7.931710.  Of  699.  A. 
8.874809.  Of  686.  A.  8.819447.  Of  880.  A.  9.604569. 
Of  981.     A.  9.936261. 

The  cube  root  of  ?i  fraction,  is  obtained  by  extracting 
the  root  of  numerator  and  denominator,  but  if  this  cannot 
be  done,  it  may  be  changed  to  a  decimal,  and  the  root  ex- 
tracted. 

Find  the  cube  root  of  y||p.     A.  ■^\.     Of  ^||ff.     A. 

2  4  Of    4  5.05.33  A         7.2  Of    'L.3_0J_3_8_4_  A"     lii.  Of 

3  1'         ^       ■5'7  0299*        ■^'     9  9*  '  '2  6  7  3  0  8  9  9  '  ^^'     2  9  9" 
2  0.3.4  C.4_1X_         _\_     213._ 

" 'Find' the  cube  Voot  of  f.  A.  .8549879.  Of  ^^V-  '^' 
.5593445.  Of  J2_4_.  A.  .4578857.  Of  ^\%.  A. 
.4562903.     Oflil.     A.  .9973262. 


ARITHMETICAL  PROGRE.S'SION. 

Any  rank,  or  series  of  numbers,  consisting  of  more  than 
two  terms,  which  increases  or  decreases  by  a  common  dif- 
ference, is  called  an  Arithmetical  series,  or  progression. 

When  the  series  increases,  that  is,  when  it  is  formed  by 
the  constant  addition  of  the  common  difference,  it  is  called 
an  ascending  series,  thus, 

1,     3,     5,     7,     9,     11,  &c. 

Here  it  will  be  seen  that  the  series  is  formed  by  a  con- 
tinual addition  of  2  to  each  succeeding  figure. 

When  the  series  decreases,  that  is,  when  it  is  formed  by 
the  constant  subtraction  of  the  common  difference,  it  is 
called  a  descending  series,  thus, 

14,     12,     10,     8,     6,     4,  &c. 

Here  the  series  is  formed  by  a  continual  subtraction  of 
2,  from  each  preceding  figure. 

The  figures  that  make  up  the  series  are  called  the 
terms  of  the  series.  The  first  and  last  terms  are  called 
the  extremes,  and  the  other  terms,  the  means. 

From  the  above,  it  may  be  seen,  that  any  term  in  a  se- 
ries may  be  found  by  continued  addition  or  subtraction, 
but  in  a  long  series  this  process  would  be  tedious.  A  much 
more  expeditious  method  may  be  found. 

1.  The  ages  o^  six  persons  are  in  arithmetical  progres- 
sion.    The  youngest  is  8  years  old,  and  the  common  dif- 


ARITHMETICAL    PROGRESSION.  249 

ference  is  3,  what  is  the  age  of  the  eldest  ?  In  other  words, 
what  is  the  last  term  of  an  arithmetical  series,  whose 
first  term  is  8,  the  number  of  terms  6,  and  the  common 
difference  3  ? 

8,     11,     14,     17,     20,     23. 

Examining  this  series,  we  find  tliat  the  common  differ- 
ence, 3,  is  added  5  times,  that  is  one  less  than  the  number 
of  terms,  and  the  last  term,  23,  is  larger  than  the  first 
term,  by  five  times  the  addition  of  the  common  difference, 
three  ;  Hence  the  age  of  the  elder  person  is  8+3x5=23. 

Therefore  when  the  first  term,  the  number  of  terms, 
and  the  common  difference,  are  given,  to  find  the  last 
term, 

Multiply  the  common  difference  into  the  number  of  terms, 
less  1,  and  add  the  product  to  the  first  term. 

2.  If  the  first  term  be.  4,  the  common  difference  3,  and 
the  number  of  terras  100,  what  is  the  last  term  ? 

Ans.  301. 

3.  There  are,  in  a  certain  triangular  field,  41  rows  of 
corn  :  the  first  row,  in  1  corner,  is  a  single  hill,  the  second 
contains  3  hills,  and  so  on,  with  a  common  difference  of  2  , 
what  is  the  number  of  hills  in  the  last  row  ?     A.  81  hills 

4.  A  man  puts  out  81,  at  6  per  cent,  simple  interest, 
which,  in  1  year,  amounts  to  81,06  in  2  years  to  81,12, 
and  so  on,  in  arithmetical  progression,  with  a  common 
difference  of  80  ,06  ;  what  would  be  the  amount  in  40 
years  ?  A.  83  ,40. 

Hence  we  see,  that  the  yearly  amomits  of  any  sum,  at 
simple  interest,  form  an  arithmetical  series,  of  which  the 
principal  is  the  first  term,  the  last  amount  is  the  last  term, 
the  yearly  interest  is  the  common  diference,  and  the  num- 
ber of  years  is  1  less  than  the  number  of  terms. 

It  is  often  necessary  to  find  the  sum  of  all  the  terms,  in 
an  arithmetical  progression.  The  most  natural  mode  of 
obtaining  the  amount  would  be  to  add  them  together,  but 
an  easier  method  may  be  discovered,  by  attending  to  the 
following  explanation. 

1.  Suppose  we  are  required  to  find  the  sum  of  all  the 
terms,  in  a  series,  whose  first  term  is  2,  the  number  of 
terms  10,  and  the  common  difference  2. 


250  ARITHMETIC.       PART  THIRD, 

2,       4,       6,       8,     10,     12,     14,     16,     18,     20 
20,     18,     16,     14,     12,     10,       8,       6,       4,       2 

22,     22,     22,     22,     22,     22,     22,     22,     22,     22^ 

The  first  row  of  figures  above,  represents  the  given  se- 
ries. The  second,  the  same  series  with  the  order  inverted, 
and  the  third,  the  sums  of  the  additions  of  the  correspond- 
ing terms  in  the  two  series.  Examining  these  series,  we 
shall  find  that  the  sums  of  the  corresponding  terms  are  the 
same,  and  that  each  of  them  is  equal  to  the  sum  of  the  ex- 
tremes, viz.  22.  Now  as  there  are  10  of  these  pairs  in 
the  two  series,  the  sum  of  the  terms  in  loth,  must  be  22x 
10=220. 

But  it  is  evident,  that  the  sum  of  the  terms  in  one  series, 
can  be  only  half  as  great  as  the  sum  of  both,  therefore,  if 
we  divide  220  by  2,  we  shall  find  the  sum  of  the  terms  in 
one  series,  which  was  the  thing  required.  220-^-2=110, 
the  sum  of  the  given  series. 

From  this  illustration  we  derive  the  following  rule  ; 

When  the  extremes  and  number  of  terms  are  given,  to 
find  the  sum  of  the  terms. 

Multiply  the  sum  of  the  extremes  by  the  number  of  terms, 
and  divide  the  product  by  2. 

2.  The  first  term  of  a  series  is  1,  the  last  term  29,  and 
the  number  of  terms  14.     What  is  the  sum  of  the  series  ? 

A.  210. 

3.  1st  term,  2,  last  term,  51,  number  of  terms,  18.  Re- 
quired the  sum  of  the  series.  A.  477. 

4.  Find  the  sum  of  the  natural  terms  1,  2,  3,  &c.  to 
10,000.  A.  50,005,000. 

5.  A  man  rents  a  house  for  ^50,  annually,  to  be  paid 
at  the  close  of  each  year  ;  what  will  the  rent  amount  to  in 
20  years,  allowing  6  per  cent.,  simple  interest,  for  the  use 
of  the  money  ? 

The  last  year's  rent  will  evidently  be  $50  without  in- 
terest,  the  last  but  one  will  be  the  amount  of  $50  for  1 
year,  the  last  but  two  the  amount  of  $50  for  2  years,  and 
so  on,  in  arithmetical  series,  to  the  first,  which  will  be  the 
amount  of  $.50  for  19  years  =  $107. 

If  the  first  term  be  50,  the  last  term  107,  and  the  num- 
ber of  terms  20,  what  is  the  sum  of  the  series  ?     A.  1570. 


ABITHMETICAL  FKOGRESSIO^i.  251 

6.  What  is  the  amount  of  an  annual  pension  of  $100, 
being  in  arrears,  that  is,  remaining  unpaid,  for  40  years, 
allowing  5  per  cent,  simple  interest  1  A.  $7900. 

7.  There  are,  in  a  certain  triangular  field,  41  rows  of 
corn  ;  the  first  row,  being  in  1  corner,  is  a  single  hill,  and 
the  last  row,  on  the  side  opposite,  contains  81  hills  ;  how 
many  hills  of  corn  in  the  field?  A.   1681. 

The  method  of  finding  the  common  difference,  may  be 
learned  by  what  follows. 

1.  A  man  bought  100  yards  of  cloth  in  Arithmetical 
progression  :  for  the  first  yard  he  gave  4  cents,  and  for  the 
last  301  cents,  what  is  the  common  increase  on  the  price 
of  each  yard  ? 

As  he  bought  100  yards,  and  at  an  increased  price  upon 
every  yard,  it  is  evident  that  this  increase  was  made  99 
times,  or  once  less  than  the  number  of  terms  in  the  series. 
Hence  the  price  of  the  last  yard  was  greater  tiian  the  first, 
by  the  addition  of  99  times  the  regular  increase. 

Therefore  if  the  first  price  be  subtracted  from  the  last, 
and  the  remainder  be  divided  by  the  number  of  additions 
(99),  the  quotient  will  be  the  common  increase  ;  301 — 4= 
297  and  297-^-99  =  3,  the  common  difference. 

Hence,  when  the  extremes  and  number  of  terms  are 
given,  to  find  the  common  difference. 

Divide  the  difference  of  the  extremes,  by  the  numher  of 
terms  less  1 . 

2.  Extremes  3  and  19  ;  number  of  terms  9.  Required 
the  common  difference.  A.   2. 

3.  Extremes  4  and  56  ;  number  of  terms  14.  Required 
the  common  difference.  A.  4. 

4.  A  man  had  15  houses,  increasing  equally  in  value, 
from  the  first,  worth  $700,  to  the  15th,  worth  $3500. 
^\'hat  was  the  difference  in  value  between  the  first  and 
second?  A.  200. 

In  Arithmetical  progression,  any  thi'ee  of  the  following 
terms  being  given,  the  other  two  maybe  found.  1.  The 
rirst  term.  2.  The  last  term.  3.  The  number  of  terms. 
4.  The  common  difference.  5.  The  sum  of  all  the 
terms. 


252  AKITHMETIC.      PART  THIRD. 

GEOMETRICAL  PROGRESSION. 

Any  series  of  numbers,  consisting  of  more  than  two 
terms,  whicli  increases  by  a  common  multiplier,  or  decrea- 
ses by  a  common  divisor,  is  called  a  Geometrical  Series. 

Thus  the  series  2,  4,  8,  16,  32,  &c.  consists  of 
terms,  each  of  which  is  tJvice  the  preceding,  and  this  is 
an  increasing  or  ascending  Geometrical  series. 

The  series  32,  16,  8,  4,  2,  consists  of  numbers, 
each  of  which  is  one  half  the  preceding,  and  this  is  a 
decreasing  or  descending  Geometrical  series. 

The  common  multiplier  or  divisor  is  called  the  Ratio, 

and  the  numbers  which  form  the  series  are  called  Terms. 

As  in  Arithmetical,  so  in  Geometrical  progression,  /if 

any  three  of  the  five  following  terms  be  given,  the  other 

two  may  be  found. 

1.  The  first  term.  2.  The  last  term.  8.  The  number 
of  terms.  4.  The  common  difference.  5.  The  sum  of 
all  the  terms. 

1.  A  man  bought  a  piece  of  cloth  containing  12  yards, 
the  first  yard  cost  3  cents,  the  second  6,  the  third  12,  and 
so  on,  doubling  the  price  to  the  last  ;  what  cost  the  last 
vard  1 

"3x2x2x'2x2x2x2x2x2x2x2x2=3X2'  '=6144,  Ans. 
In  examining  the  above  process,  it  will  be  seen,  that  the 
price  of  the  second  yard  is  found  by  multiplying  the  first 
payment  into  the  ratio  (2)  once  ;  the  price  of  the  third 
yard,  by  multiplying  by  2  ticice,&c.,  and  that  the  ratio  (2) 
is  used  as  a  tactor  eleven  times,  or  once  less  than  the  num. 
ber  of  terms.  The  last  term  then,  is  the  eleventh  power  of 
the  ratio  (2)  multiplied  by  the  first  term  (3). 

Hence  the  first  term,  ratio,  and  number  of  terms,  being 
given,  to  find  the  last  term. 

Midtiply  the  first  term,  by  that  poiver  oj  the  ratio,  whose 
index  is  one  less  than  the  number  of  terms. 

Note.  In  involving  the  ratio,  it  is  not  always  necessary 
to  produce  all  the  intermediate  powers  ;  the  process  may 
often  be  abridged,  by  multiplying  together  two  powers  al- 
ready obtained,  thus, 

The  11th  power  =  the  6th  power  X  the  5th  power,  dec. 


GEOMETRICAL  PROGRESSION.  253 

2.  If  the  first  terra  is  2,  the  ratio  2,  and  the  number  of 
terms  13,  what  is  the  last  term  ?  A.  8,192. 

3.  Find  the  12th  term  of  a  series,  whose  first  term  is  3, 
and  ratio,  3.  A.  531,441. 

4.  A  man  plants  4  kernels  of  corn,  which,  at  harvest, 
produce  32  kernels  ;  these  he  plants  the  second  year ; 
now,  supposing  the  annual  increase  to  continue  8  fold, 
what  would  be  the  produce  of  the  16th  year,  allowing  1000 
kernels  to  a  pint?  A.  2199023255.552  bushels. 

5.  Suppose  a  man  had  put  out  one  cent  at  compound 
interest  in  1620,  what  would  have  been  the  amount  in 
1824,  allowing  it  to  double  once  in  12  years  ? 

217=131072.  A.  1310.72. 

The  most  obvious  method  of  obtaining  the  sum  oftlie  terms 
in  a  Geometrical  series,  might  be  by  addition,  but  this  is 
not  the  most  expeditious,  as  will  be  seen. 

1.  A  man  bought  5  yards  of  cloth,  giving  2  cents  for  the 
first,  6  cents  for  the  second,  and  so  in  3  fold  ratio  ;  what 
did  the  whole  cost  him  ? 

2,     6,     18,     54,     162 

6,     18,     54,     162,     486 

The  first  of  the  above  lines,  represents  the  original 
series.     The  second,  that  series,  multiplied  by  the  ratio  3. 

Examining  these  series,  it  will  be  seen  that  their  terms 
are  all  alike  excepting  two  :  viz.  i\i(i  first  term  of  the  first 
series,  and  the  last  of  the  second  series.  If  now  we  sub- 
tract  the  first  series  from  the  last,  we  have  for  a  remainder 
486 — 2=484,  as  all  the  intermediate  terms  vanish  in  the 
subtraction. 

Now  the  last  series  is  three  times  the  first,  (for  it  was 
made  by  multiplying  the  first  series  by  3,)  and  as  we  have 
already  subtracted  once  the  first,  the  remainder  must  of 
course  be  twice  the  first. 

Therefore  if  we  divide  484  by  2,  we  shall  obtain  the 
sum  of  the  first  series.  484-f-2=242    Ans. 

As  in  the  preceding  process,  all  the  terms  vanish  in  the 
subtraction,  excepting  the  first  and  last,  it  will  be  seen, 
that  the  result  would  have  been  the  same,  if  the  last  term 
only,  had  been  multiplied,  and  the  first  subtracted  from 
the  product. 

22 


254  ARITHMETIC.       FART  THIRD. 

Hence,  the  extremes  and  ratio  being  given,  to  find  the 
sum  of  all  the  terms. 

Multiply  the  greater  term  by  the  ratio,  from  the  product 
subtract  the  least  term,  and  divide  the  remainder  by  the  ratio 
less  1. 

2.  Given  the  first  term,  1  ;  the  last  term,  2,187  ;  and 
the  ratio,  3  ;   required  the  sum  of  the  series.     A.  3,280. 

3.  Extremes,  1  and  65,536;  ratio  4  ;  required  the  sum 
of  the  series.     A.  87,381. 

4.  Extremes,  1,024  and  59,049  ;    required   as  above. 

A.   175,099. 

5.  What  is  the  sum  of  the  series  16,  4,  1,  I,  j\,  ^'j, 
and  so  on,  to  an  infinite  extent  ?  A.  21^. 

Here  it  is  evident,  the  last  term  is  0,  or  indefinitely  near 
to  nothing,  the  extremes  therefore  are  16  and  0,  and  the 
ratio  4. 


ANNUITIES. 

An  annuity  is  a  sum  payable  periodically,  for  a  certain 
length  of  time,  or  forever. 

An  annuity,  in  the  proper  sense  of  the  word,  is  a  sum 
paid  annually,  yet  payments  made  at  different  periods,  are 
called  annuities.  Pensions,  rents,  salaries,  &c.  belong  to 
annuities. 

When  annuities  are  not  paid  at  the  time  they  become 
due,  they  are  said  to  be  in  arrears. 

The  sum  of  all  the  annuities  in  arrears,  with  the  interest 
on  each  for  the  time  they  have  remained  due,  is  called  the 
amount. 

The  Present  worth  of  an  annuity,  is  the  sum  which  should 
be  paid  for  an  annuity  yet  to  come. 

When  an  annuity  is  to  continue  forever,  its  present 
worth  is  a  sum,  whose  yearly  interest  equals  the  annuity. 

Now  as  the  principal,  multiplied  by  the  rate,  will  give 
the  interest,  the  interest,  divided  by  the  rate,  will  give  the 
principal. 

Hence  to  find  the  present  worth  of  an  annuity,  continuing 
forever, 

Divide  the  annuity  by  the  rate  per  cent. 


ANNUITIES.  255 

1.  What  is  the  worth  of  -$100  annuity,  to  continue  for- 
ever,  allowing  to  the  purchaser  4  per  cent.  ?  allowing  5 
per  cent.  ?  8  per  cent.  ?  10  per  cent.  ?  15  per  cent.  ? 
20  per  cent.  ?  Ans.  to  last,  .$500. 

2.  What  is  an  estate  worth,  which  brings  in  $7,500  a 
year,  allowing  G  per  cent.  ?  A.  $125,000. 


ANNUITIES  AT  COMPOUND  INTEREST. 

It  has  been  shown  (page  208)  that  Compound  Interest 
is  that  which  arises  from  adding  the  interest  to  the  principal 
at  the  close  of  each  year,  and  making  the  amount  a  new 
a  new  principal.  The  amount  of  $1  for  one  year  at  6 
percent,  is  $1.06,  and  it  will  be  found,  that  iftlie  princi- 
pal be  multi|)lied  by  this,  the  product  will  be  the  amount 
for  I  year,  and  this  amount  multiplied  by  1.0(5,  will  be  the 
amount  for  2  years,  and  so  on.  Hence  we  see  that  any 
sum  at  compound  interest,  forms  a  geometrical  series,  of 
which  the  ralio  is  the  amount  of  $1  at  the  given  rate  per 
cent. 

1.  An  annuity  of  $40  was  left  5  years  unpaid,  what  was 
then  due  upon  it,  allowing  5  per  cent,  compound  interst  ? 

It  is  evident  that  for  the  fifth  or  last  year,  the  armuity 
alone  is  due  ;  i'oi'  i\\Q  fourth,  the  amount  of  the  annuity  for 
1  year  ;  for  the' third  the  amount  of  the  annuity  for  2  years, 
and  so  on  ;  and  the  sum  of  these  amounts  will  be  the 
answer,  or  what  is  due  in  5  years. 

'From  this  we  find  that  the  amount  of  an  annuity  in  ar- 
rears, fornas  a  geometrical  progression,  whose^r6*<  term  is 
the  annuity,  the  ratio,  the  amount  of  $1  at  the  given  rate, 
and  the  number  of  terms,  the  number  of  years. 

The  above  example,  then,  may  be  resolved  into  the  fol- 
lowing question.  What  is  the  sum  of  a  geometrical  series 
whose  first  term  is  ,^*40,  the  ratio  1.05,  and  the  number  of 
terms  5  1  First  find  the  last  term,  by  the  first  rule  in  Geo- 
metrical  progression,  and  then  the  sum  of  the  series  by 
the  second  rule.    The  answer  will  be  found  to  be  $221.02. 

Hence,  to  find  the  amount  of  an  annuity  in  arrears,  at 
compound  interest. 

Find  the  sum  of  a  Geometrical  series,  whose  first  term  is 


256  ARITHMETIC.       PART   THIRD. 

the  annuity,  whose  ratio,  the  amount  of  ^l  at  the  given  rate 
per  cent.,  and  whose  number  of  terms  is  the  number  of  years. 

Note.  A  table,  showing  the  amount  of  $1  at  5  and  6 
per  cent.,  compound  interest,  for  any  number  of  years  not 
exceeding  24,  will  be  found  on  page  209. 

2.  What  is  the  amount  of  an  annuity  of  $50,  it  being 
in  arrears  20  years,  allowing  5  per  cent,  compound  inte- 
rest? A.  81653,29. 

3.  If  the  annual  rent  of  a  house,  which  is  $150,  be  in 
arrears  4  years,  what  is  the  amount,  allowing  10  per  cent, 
compound  interest  ?  A.  $696,15. 

4.  To  how  much  would  a  salary  of  $500  per  annum 
amount  in  14  years,  the  money  being  improved  at  6  per 
cent.,  compound  interest  ?  in  10  years  ?  in  20  years  ? 
in  22  years  ?     in  24  years  ? 

Ans.  to  the  last,  $25,407,75. 

5.  Find  the  amount  of  an  annuity  of  $159,  for  3  years, 
at  6  per  cent.  A.  $477,54. 

A  rule  has  been  given,  for  finding  the  present  worth  of 
an  annuity,  to  continue  forever  ;  but  it  is  often  necessary 
to  find  the  present  worth  of  an  annuity,  which  is  to  con- 
tinue for  a  limited  number  of  years  ;  thus, 

6.  What  is  the  present  worth  of  an  annual  pension  of 
$100  to  continue  4  years,  allowing  6  per  cent,  compound 
interest? 

The  present  worth  is  evidently  a  sum,  which,  at  com- 
pound  interest,  would  in  4  years  produce  an  amount  equal 
to  the  amount  of  the  annuity,  for  the  same  time. 

Now  to  find  a  given  amount,  at  compound  interest,  we 
multiply  a  sum  by  the  amount  of  $1  at  the  given  rate  per 
cent,  as  many  times  successively  as  there  are  years. 

Hence  to  find  a  sum,  which  will  produce  a  given 
amount  in  a  certain  time,  we  must  reverse  this  process  and 
divide  by  the  amount  of  ,^1  for  the  given  time. 

Applying  this  to  the  above  example,  we  find  by  the  pre- 
ceding rule,  that  the  amount  is  $437,46  Dividing  this  by 
the  amount  of  $1  for  4  years,  we  find  the  present  worth, 
437,46-r  1,26247  =$346,511,  Ans, 


ANNUITIES. 


257 


Heace  to  find  the  present  worth  of  an  annuity, 

Find  the  amount  in  arrears  jor  the  whole  time,  and  divide 
it  by  the  amount  oj  $i  at  the  given  rate  per  cent.,  for  the 
given  number  of  years. 

The  operations  under  this  rule,  will  be  facilitated  by  the 
following 

TABLE, 

showing  the  present  worth  of  $1,  or  £1  annuity,  at  5  and 
6  per  cent,  compound  interest,  for  any  number  of  years 
from  1  to  34. 


Years.    5  per  cent. 


1 

0,95238 

2 

1,85941 

3 

2,72325 

4 

3,54595 

5 

4,32948 

6 

5,07569 

7 

5,78037 

8 

6,46321 

9 

7,10782 

10 

7,72173 

11 

8,306U 

12 

8,86325 

13 

9,39357 

14 

9,89864 

15 

10,37966 

16 

10,83777 

17 

11,27407 

6  per  cent. 

0,94339 
1,83339 
2,67301 
3,4651 
4,21236 
4,91732 
5,58238 
6,20979 
6,80169 
7,36008 
7,88687 
8,38384 
8,85268 
9,29498 
9,71225 
10,10589 
10,47726 


Years. 

18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 


5  per  cent. 

11,68958 

12,08532 

12,46221 

12,82115 

13,163 

13,48807 

13,79864 

14,09394 

14,37518 

14,64303 

14,89813 

15,14107 

15,37245 

15,59281 

15,80268 

16,00255 

16,1929 


6  per  cent. 

10,8276 

11,15811 

11,46992 

11,76407 

12,04158 

12,30338 

12,55035 

12,78335 

13,00316 

13,21053 

13,40616 

13,59072 

13,76483 

13,^.2908 

14,08398 

14,22917 

14,36613 


It  is  evident,  that  the  present  worth  of  $2  annuity  is  2 
times  as  much  as  that  of  $1  ;  the  present  worth  of  ^3  will 
be  3  times  as  much,  &c.  Hence,  to  find  the  present  worth 
of  any  annuity,  at  5  or  6  per  cent, — Find,  in  this  table,  the 
present  worth  of  il  annuity,  and  multiply  it  by  the  given 
annuity,  and  the  product  will  be  the  present  worth. 

7.  Find  the  present  worth  of  a  $40  annuity,  to  continue 
5  years,  at  5  per  cent.  A.  $173,173. 

8.  Find  the  present  worth  of  $100  annuity,  for  20  years, 
at  5  per  cent.  A.  $1,246.22. 

9.  Find  the  present  worth  of  an  annuity  of  $21,54  for 
7  years  at  6  per  cent.  A.  120.2444- 

22* 


259  ARITHMETIC.        PART   THIRD. 

10.  Find  the  present  worth  of  an  annuity  of  $100,  to 
continue  J  2  years,  at  6  per  cent.  A.  $838,384. 

11.  Find  the  present  worth  of  an  annuity  of  $936,  for 
20  years,  at  5  per  cent.  A.  $1 1,664.629 — 

As  the  present  worth  of  any  annuity  may  be  found,  by 
multiplying  the  annuity  by  one  of  the  numbers,  in  the 
above  table,  it  is  plain  that  if  any  present  worth  be  divided 
by  the  same  number,  it  will  give  the  annuity  itself. 

Hence  to  discover  of  what  annuity  any  given  sum  is  the 
present  worth,  we  may  use  the  above,  as  a  table  of  divi- 
sors, instead  of  multipliers. 

What  annuity  to  continue  19  years,  will  $0,094,866 
purchase,  when  money  will  bring  6  per  cent.  ?       A.  ^600. 

An  annuity  is  said  to  be  in  reversion,  when  it  does  not 
commence  until  some  future  time. 

12.  What  is  the  present  worth  of  $60  annuity,  to  be  con- 
tinued  6  years,  but  not  to  commence  till  3  years  hence, 
allowing  6  per  cjent.  compound  interest  ? 

The  present  worth  is  evidently  such  a  sum  as  would  in 
3  years,  at  six  per  cent.,  compound  interest,  produce  an 
amount,  equal  to  the  present  worth  of  the  annuity,  were  it 
to  commence  immediately. 

We  must  therefore  first  find  the  present  worth  of  an  an- 
nuity of  $60  to  commence  immediately,  according  to  the 
last  rule.     This  we  shall  discover  to  be  $295,039. 

We  now  wish  to  obtain  a  sum,  whose  amount  in  3  years 
will  equal  this  present  worth.     This  may  be  found  by  di- 
viding the  $295,039  by  the  amount  of  $1  for  3  years  thus, 
$295,039-^1,19101=247.72. 

Ans.  $247,72. 

Hence  to  find  the  present  worth  of  any  annuity  taken  in 
reversion,  at  compound  interest. 

Find  the  present  worth  to  commence  immediately,  and  this 
sum  divided  by  the  amount  q/  $1  for  the  time  in  reversion, 
will  give  the  answer. 

13.  If  an  annuity  of  $100  be  14  years  in  reversion,  to 
continue  20  years  afterwards,  what  is  its  present  worth, 
discounting  at  5  per  cent.  ?  A.  ^629.426. 

14.  What  is  the  present  worth  of  a  lease  of  $100  to 
continue  20  years,  but  not  to  commence  till  the  end  of  4 


PERMUTATION.  259 

years,  allowing  5  per  cent.  ?    what  if  it  be  6  years  in  re- 
version ?     8  years  ?     10  years  ?     14  years  ? 

Ans.  to  last,  $629,426. 

15.  What  is  the  present  worth  of  $100  annuity,  to  be 

continued  4  years,  but  not  to  commence  till  2  years  hence, 

allowing  6  per  cent,  compound  interest  ?      A.  $308,393. 


PERMUTATION. 

Permutation  is  the  method  of  finding  how  many  chan- 
ges may  be  made,  in  the  order  in  which  things  succeed 
each  other. 

What  number  of  permutations  may  be  made  on  the  let- 
ters A  and  B ?     They  may  be  written  A  B,  or  B  A. 

What  number  on  the  letters  ABC? 

Placing  A  first,  A  B  C,  or  A  C  B. 

Placing  B  first,  B  A  C,  or  B  C  A. 

Placing  C  first,  C  A  B,  or  C  B  A. 

From  tliese  examples  it  will  be  seen,  that  of  two  things 
there  mav  be  2  changes,  (1x2=2,)  and  of  3  things  there 
may  be 6  changes.  (1X2X3=6.) 

Hence,  to  find  the  number  of  different  changes,  or  per- 
mutations, of  which  any  number  of  different  things  are 
capable. 

Find  the  continual  product  of  the  natural  series  of  num- 
bers, from  1  to  the  given  number. 

1.  Four  gentlemen  agreed  to  remain  together,'  as  long 
as  they  could  arrange  themselves  differently  at  dinner. 
How  many  days  did  they  remain  ?  A.  24  days. 

2.  10  gentlemen  made  the  same  agreement,  but  they 
all  died  before  it  could  be  fulfilled.  The  last  survivor  lived 
53  yrs.  98  days,  after  the  agreement.  How  much  did  the 
bargain  then  want  of  being  fulfilled,  allowing  365  days  to 
the  year  1  A.  9,888  yrs.  237  d. 

3.  How  many  years  will  it  take  to  ring  all  the  possible 
changes  on  12  bells,  supposing  that  10  can  be  rung  in  a 
minute,  and  that  the  year  contains  365  d.  5  h.  49  m  ? 

A.  91  yrs.  20  d.  22  h.  41m. 

4.  How  many  variations  may  there  be  in  the  position  of 
the  nine  digits  J  Ans.  362880 


260  ARITHMETIC.       PART  THIRD. 

5.  A  man  bought  25  cows,  agreeing  to  pay  for  them  1 
cent  for  every  different  order  in  which  they  could  all  be 
placed  ;  how  much  did  the  cows  cost  him  ? 

Ans.  il551 12100433309859840000. 


MISCELLANEOUS  EXAMPLES. 

Many  of  these  sums  are  designed  for  mental  exercise. 
In  solving  the  first  50,  the  pupil  should  not  be  allowed  to 
use  the  slate. 

1.  If  two  men  start  from  the  same  place  and  travel  in 
opposite  directions,  one  at  the  rate  of  4|  miles  an  hour, 
and  the  other  at  the  rate'of  3|  miles  an  hour,  how  far  will 
they  be  apart  in  6  hours  ? 

2.  If  6  bushels  of  oats  will  keep  3  horses  a  week,  how 
many))ushels  will  be  required  to  keep  12  horses  the  same 
time  1 

3.  If  you  give  5  men  3f  bushels  of  corn  apiece,  how 
much  do  you  give  the  whole  ? 

4.  If  8  dollars  worth  of  provisions  will  serve  9  men  5 
days,  how  many  days  will  it  serve  12  men  ?  how  many 
days  would  it  serve  3  men  ? 

5.  If  86  worth  of  provision  will  serve  5  men  8  days, 
how  many  days  would  it  serve  9  men  1  how  many  days 
would  it  serve  3  men  ? 

6.  If  $12  worth  of  provision  would  serve  5  men  7  days, 
how  many  men  would  it  serve  9  days  ? 

7.  If  one  peck  of  wheat  afford  9  six  penny  loaves,  how 
many  ten  penny  loaves  would  it  afford  ? 

8.  If  a  man  paid  $60  to  his  laborers,  giving  to  every 
man  9d.  and  to  every  boy  3d.  if  the  men  and  boys  were 
equal  in  number,  how  many  were  there  of  each  ? 

9.  Two  men  bought  a  barrel  of  flour  together,  one  paid 
$3  and  the  other  paid  $5  ;  what  part  of  the  whole  did  each 
pay,  and  what  part  of  the  barrel  ought  each  to  have  ? 

10.  Three  men  hired  a  field  together,  A  paid  $7,  B 
paid  ^3,  and  C  paid  $8,  what  part  of  the  whole  did  each 
pay,  and  what  part  of  the  produce  ought  each  to  have  ? 

11.  Three  men  bought  a  lottery  ticket  together,  A  paid 
^6,  B  paid  $4,  and  C  paid  $10.  They  drew  a  prize  of 
$150,  what  was  each  man's  share  1 


MISCELLANEOUS  EXAMPLES.  261 

12.  Three  men  hired  a  pasture  together  for  $60.  A 
put  in  2  horses,  B  4  horses,  and  C  6  horses,  how  much 
ought  each  to  pay  ? 

13.  Three  men  commenced  trade  together,  and  ad- 
vanced  money  in  this  proportion — For  every  $5  that  A 
put  in,  B  put  in  3,  and  C  put  in  $2,  they  gained  $100, 
what  was  each  man's  share  ? 

14.  Two  men  hired  a  pasture  for  ^32.  A  put  in  3 
sheep  for  4  months,  and  B  put  in  4  sheep  for  5  months, 
how  much  ought  each  to  pay  ? 

Note.  3  sheep  for  4  months  is  the  same  as  12  sheep 
for  one  month,  and  4  sheep  for  5  months  is  the  same  as  20 
sheep  for  one  month. 

15.  A  and  B  traded  together  and  invested  money  in  the 
following  proportions,  A  put  in  10  for  2  months,  and  B 
put  in  $5  for  3  months.  They  gained  $70  ;  what  was 
each  man's  share  ? 

16.  Three  men  traded  in  company,  and  put  in  money  in 
the  following  proportions.  A  put  in  4  dollars  as  often  as 
B  put  in  3,  and  as  often  as  C  put  in  2.  A's  money  was  in 
2  months,  B's  3  months,  and  C's  4  months.  They  gained 
$100  ;  what  was  each  man's  share  ? 

17.  Two  men  traded  in  company.  A  put  in  $2  as  oflen 
as  B  put  in  $3.  A's  money  was  employed  7  months,  and 
B's  5  months.  They  gained  58  dollars.  What  was  each 
man's  share  ? 

18.  If  A  can  do  i  of  a  piece  of  work  in  1  day,  and  B 
can  do  ^  of  it  in  one  day,  how  much  would  both  do  in  a 
day  ?  How  long  would  it  take  them  both  together  to  do  the 
whole  ? 

19.  If  1  man  can  do  a  piece  of  work  in  2  days,  and 
another  in  3  d'ays,  how  much  of  it  would  each  do  in  a  day  ? 
How  much  would  both  together  do  ?  How  long  would  it 
take  them  both  to  do  the  whole  1 

20.  A  cistern  has  2  cocks ;  the  first  will  fill  it  in  3 
hours,  the  second  in  6  hours  ;  how  much  of  it  would  each 
fill  in  an  hour  ?  How  much  would  both  together  fill  ?  How 
long  would  it  take  them  both  to  fill  it  1 

21.  A  man  and  his  wife  found  by  experience,  that,  when 
when  they  were  both  together,  a  bushel  of  meal  would 
last  them  only  2  weeks ;  but  when  the  man  was  gone,  it 


262  ARITHMETIC.       PART  THIRD. 

would  last  his  wife  5  weeks.  How  much  of  it  did  both 
together  consume  in  1  week  ?  What  part  did  the  wonaan 
alone  consume  in  1  week  ?  What  part  did  the  man  alone 
consume  in  1  week  ?  How  long  would  it  last  the  man 
alone  ? 

22.  If  1  man  could  build  a  piece  of  wall  in  5  days,  and 
another  man  could  do  it  in  7  days,  how  much  of  it  would 
each  do  in  1  day  ?  How  many  days  would  it  take  them 
both  to  do  it  ? 

23.  A  cistern  has  3  cocks  ;  the  first  would  fill  it  in  3 
hours,  the  second  in  6  hours  ;  the  third  in  4  hours  ;  what 
part  of  the  whole  would  each  fill  in  1  hour  ?  and  how  long 
would  it  take  them  all  to  fill  it,  if  they  were  all  running  at 
once? 

24.  A  and  B  together  can  build  a  boat  in  8  days,  and 
with  the  assistance  of  C  they  can  do  it  in  5  days ;  how 
much  of  it  can  A  and  B  build  in  1  day?  How  much  of  it 
can  A,  B,  and  C,  build  in  1  day?  How  much  of  it  can  C 
build  alone  in  1  day  ?  How  long  would  it  take  C  to  build  it 
alone  ? 

25.  Suppose  T  would  line  8  yards  of  broadcloth  that  is 
li  yard*  wide,  with  shalloon  that  is  |  of  a  yard  wide  ;  how 
many  yards  of  the  shalloon  will  line  1  yard  of  the  broad- 
cloth 1  How  many  yards  will  line  the  whole  ? 

26.  If  7  yards  of  cloth  cost  13  dollars,  what  will  10 
yards  cost  ? 

27.  If  the  wages  of  25  weeks  come  to  75  dollars,  what 
will  be  the  wages  of  seven  weeks  ? 

28.  If  8  tons  of  hay  will  keep  7  horses  three  months, 
how  much  will  keep  12  horses  the  same  time  ? 

29.  If  a  stafr4  feet  long  cast  a  shadow  6  feet  long,  what 
is  the  length  of  a  pole  that  casts  a  shadow  58  feet  at'  the 
same  time  of  day  ? 

30.  If  a  stick  8  feet  long  cast  a  shadow  2  feet  in  length, 
what  is  the  height  of  a  tree  which  casts  a  shadow  42  feet 
at  the  same  time  of  day  ? 

31.  A  ship  has  sailed  24  miles  in  4  hours  ;  how  long 
will  it  take  her  to  sail  150  at  the  same  rate  ? 

32.  30  men  can  perform  a  piece  of  work  fn  20  days  ; 
how  many  men  will  it  take  to  perform  the  same  work  in 
8  days  ? 


MISCELLANEOUS  EXAMPLES.  263 

33.  17  men  can  perform  a  piece  of  work  in  25  days  ; 
in  how  many  days  would  5  men  perform  ihe  same  work  ? 

34.  A  hare  has  76  rods  the  start  of  a  greyhound,  but 
the  greyhound  runs  15  rods  to  10  of  the  hare  ;  how  many 
rods  must  the  greyhound  run  to  overtake  the  hare  ? 

35.  A  garrison  has  provision  for  8  months,  at  the  rate 
of  15  ounces  per  day ;  how  much  must  be  allowed  per 
day,  in  order  that  the  provision  may  last  1 1  months  ? 

36.  If  8  men  can  build  a  wall  15  rods  in  length  in  10 
days,  how  many  men  will  it  take  to  build  a  wall  45  rods 
in  length  in  5  days? 

37.  A  man  being  asked  the  price  of  his  horse,  an- 
swered, that  his  horse  and  saddle  together  were  worth 
100  dollars  ;  but  the  horse  was  worth  9  times  as  much  as 
the  saddle.     What  was  each  worth  1 

38.  A  man  having  a  horse,  a  cow,  and  a  sheep,  was 
Ksked  what  was  the  value  of  each.  He  answered  that  the 
cow  was  worth  twice  as  much  as  the  sheep,  and  the  horse 
3  times  as  much  as  the  sheep,  and  that  all  together  were 
worth  60  dollars.     What  was  the  value  of  each  ? 

39.  If  80  dollars  worth  of  provision  will  serve  20  men 
24  days,  how  many  days  will  100  dollars  worth  of  provi- 
sion serve  30  men  ? 

40.  The  third  part  of  an  army  was  killed,  the  fourth 
part  taken  prisoners,  and  1000  fled  ;  how  many  were  in 
this  army  ? 

This,  and  the  following  10  questions,  are  usually  classed 
under  the  rule  opposition,  bat  they  may  be  solved  in  a  much 
more  simple  and  easy  manner.  Thus,  i-(-i=_7_of  the  army. 
Now  as  there  are  12  twelfths  in  the  whole,  1000  must  be 
the  remaining  5  twelfths.  If  1000  is  5  twelfths  of  the  ar- 
my, 1  fifth  of  1000,  or  200,  will  be  1  twelfth  ;  and  if  200 
is  1  twelfth,  the  whole,  or  12  twelfths  will  be  12  times  as 
much,  or  2400. 

41.  A  farmer  being  asked  how  many  sheep  he  had,  an- 
swered, that  he  had  them  in  4  pastures  ;  in  the  first  he 
had  i  of  his  flock  ;  in  the  second  i ;  in  the  third  }  ;  and  in 
the  fourth  15  ;  how  many  sheep  had  he  ? 

42.  A  man  driving  his  geese  to  market,  was  met  by 
another,  who  said,  good  morrow,  master,  with  your  hun- 
dred  geese  ;  says  he,  I  have  not  a  hundred  ;  but  if  I  had 


264  ARITHMETIC.       PART   THIRD. 

half  as  many  more  as  I  now  have,  and  two  geese  and  a 
half,  I  should  have  a  hundred  ;  how  many  had  he  1 

43.  What  number  is  that,  to  which  if  its  half  be  added 
the  sura  will  be  CO  ? 

44.  What  number  is  that,  to  which  if  its  third  be  added 
the  sum  will  be  48  ? 

45.  What  number  is  that,  to  which  if  its  5th  be  added 
the  sum  will  be  54  ? 

46.  What  number  is  that,  to  which  if  its  half  and  its 
third  be  added  the  sum  will  be  55  ?       ; 

47.  A  man  being  asked  his  age,  answered,  that  if  its 
half  and  its  third  were  added  to  it,  the  sum  would  be  77  ; 
what  was  his  age  ?  . 

48.  What  number  is  that,  which  being  increased  by  its 
half,  its  fourth,  and  eighteen  more,  will  be  doubled? 

49.  A  boy  being  asked  his  age,  answered,  that  if  ^  and 

1  of  his  age,  and  20  more  were  added  to  his  age,  the  sum 
would  be  3  times  his  age.     What  was  his  age  ? 

50.  A  man  being  asked  how  many  sheep  he  had,  an- 
shered,  that  if  he  had  as  many  more,  ^  as  many  more, 
and  21  sheep,  he  should  have  100.     How  many  had  he  ? 

51.  A  farmer  carried  his  grain  to  market,  and  sold 
75  bushels  of  wheat,  at  $1,45  per  bushel, 

64     „  „  rye,       „  $   ,95  „       „ 

142     „  „  corn,     „  $  ,50  „       „ 

In  exchange  he  received  sundry  articles  : — 
3  pieces  of  cloth,  each 

containing  31  yds,  at  $1,75  per  yd. 

2  quintals  offish,'        „  $2,30  per  quin. 
8  hhds.  of  salt,  „  $4,30  per  hhd. 


and  the  balance  in  money. 

How  much  money  did  he  receive  ?  Ans,  $38,80 

52.  A  man  exchanges  760  gallons  of  molasses,  at  37| 
cents  per  gallon,  for  661  cwt.  of  cheese,  at  $4  per  cwt.  ; 
how  much  will  be  the  balance  in  his  favor  ?        Ans.  $19 

53.  Bought  84  yards  of  cloth,  at  $1,25  per  yard  ;  how 
much  did  it  come  to  ?  How  many  bushels  of  wheat,  at 
$1,50  per  bushel,  will  it  take  to  pay  for  it  ? 

Ans.  to  the  last,  70  bushels. 


MISCELLANEOUS  EXAMPLES.  265 

54.  A  man  sold  342  pounds  of  beef,  at  6  cents  per 
pound,  and  received  his  pay  in  molasses,  at  d7\  cents  per 
gallon  ;  how  many  gallons  did  he  receive  ? 

Ans.  54,72  gallons. 

55.  A  man  exchanged  70  bushels  of  rye,  at  8,92  per 
bushel,  for  40  bushels  of  wheat,  at  -$1,371  per  bushel,  and 
received  the  balance  in  oats,  at  f  ,40  per  bushel  ;  how 
many  bushels  of  oats  did  he  receive  ?  Ans.  23i 

56.  How  many  bushels  of  potatoes,  at  1  s.  0  d.  per 
bushel,  must  be  given  for  32  bushels  of  barley,  at  2  s.  6  d. 
per  bushel  ?  Ans  53^^  bushels. 

57.  How  much  salt,  at  -11,50  per  bushel,  must  be  given 
in  exchange  for  15  bushels  of  oats,  at  2  s.  3  d.  per  bushel  ? 

Note.  It  will  be  recollected  that,  when  the  price  and 
cost  are  given,  to  find  the  quantity,  they  must  both  be  re- 
duced  to  the  same  denomination  before  dividing. 

Ans.  3^  bushels. 

58.  How  much  wine,  at  $2,75  per  gallon,  must  be 
given  in  exchange  for  40  yards  of  cloth,  at  7  s.  6  d,  per 
yard  ?  Ans.  1  Sf^  gallons. 

59.  There  is  a  fish,  whose  head  is  4  feet  long  ;  his  tail 
is  as  long  as  his  head  and  |  the  length  of  his  body,  and  his 
body  is  as  long  as  his  head  and  tail ;  what  is  the  length  of 
the  fish  ? 

The  pupil  will  perceive  that  the  length  of  the  body  is 
i  the  length  of  the  fish.  Ans.  32  feet. 

60.  A  gentleman  had  7  £.  17  s.  6  d.  to  pay  among  his 
laborers  ;  to  every  boy  he  gave  6  d.,  to  every  woman  8d., 
and  to  every  man  16  d. ;  and  there  were  for  every  boy 
three  women,  and  for  every  woman  two  men  ;  I  demand 
the  number  of  each.  Ans.  15  boys,  45  women,  and  90men. 

61.  A  farmer  bought  a  sheep,  a  cow,  and  a  yoke  of 
oxen  for  $82,50  ;  he  gave  for  the  cow  8  times  as  much  as 
for  the  sheep,  and  for  the  oxen  3  times  as  much  as  for  the 
cow  ;  how  much  did  he  give  for  each  ? 

Ans.  For  the  sheep  $2,50,  the  cow  $20,  and  the  oxen 


62.  There  was  a  farm,  of  which  A  owned  f ,  and  B  if  ; 
the  farm  was  sold  for  $1764;  what  was  each  one's  share 
of  the  money  1  Ans.  A's  $504,  and  B's  $1260 

23 


266  ARITHMETIC.       PART   THIRD. 

63.  Four  men  traded  together  on  a  capital  of  $3000,  of 
which  A  put  in  1,6^,0  i,  and  DyL  ;  at  the  end  of  3  yrs., 
they  had  gained  $2364  ;  what  was  each  one's  share  of  the 
gain?  rA's$1182 

.        1  B's  $  591 

■^"^-  )  C's  $  394 

'  D's  $  197 

64.  Bought  a  book,  the  price  of  which  was  marked 
$4,50,  but  for  cash  the  bookseller  would  sell  it  at  33^  per 
cent,  discount ;  what  is  the  cash  price  ?  Ans.  $3,00 

65.  A  merchant  bought  a  cask  of  molasses,  containing 
120  gallons,  for  *^42  ;  for  how  much  must  he  sell  it  to  gain 
15  per  cent.  ?  How  much  per  gallon  ?    Ans.  to  last,  $,40i 

Q6.  A  merchant  bought  a  cask  of  sugar,  containing  740 

pounds,   for  $59,20 ;  how  must  he  sell  it  per  pound  to 

gain  25  per  cent  ?  Ans,  $,10 

'67.  What  is  the  interest,  at  6  per  cent.,  of  $71,02  for 

17  months  12  days  ?  Ans.  $6,178+ 

68.  What  is  the  interest  of  $487,003  for  18  months  ? 

Ans.  $43,83+ 
It  has  been  shown  that   the    length   of  one    side  of  a 
square   multiplied   into   itself,  will   give  the  square  con- 
tents. 

Hence  to  find  the  area,  or  superficial  contents  of  a 
square  when  one  side  is  given. 

Multiply  the  side  of  the  square  into  itself. 

69.  There  is  a  room  18  feet  square  ;  how  many  yards 
of  carpeting  1  vard  wide  will  cover  it  ? 

Ans.  182=324  ft.=36  yards. 

70.  The  length  of  one  side  of  a  square  room  is  31  feet; 
how  many  square  feet  in  the  whole  room  1  Ans.  961 

71.  If  the  floor  of  a  square  room  contain  36  square 
yards,  hovv  many  feet  does  it  measure  on  each  side  ? 

Ans.  18  feet. 

Note.  This  answer  is  obtained  by  finding  the  square 
root  of  the  area  36  feet. 

A  parallelogram,  or  ohlong,  is  a  four  sided  figure,  ha- 
ving its  opposite  sides  equal  and  parallel. 

To  find  the  area  of  a  parallelogram. 

Multiply  the  length  by  the  breadth. 


MISCELLANEOUS  EXAMPLES.  267 

72.  A  garden  in  the  form  of  a  parallelogram  is  96  feet 
long  and  54  wide  ;  how  many  square  feet  of  ground  are 
contained  in  it?  Ans.  5184  sq.  ft. 

73.  What  is  the  area  of  a  parallelogram  120  rods  long 
and  60  wide  ?  Ans.  7200  sq.  rods. 

74.  If  a  board  be  21  feet  long,  and  18  inches  broad, 
how  many  square  feet  are  contained  in  it  ? 

Ans.  3H  sq.  feet. 
A  triangle  is  a  figure  bounded  by  three  lines. 
If  a  line  be  drawn  from  one  corner  of  a  parallelogram 
to  its  opposite,  (as  in  the  Fig.  A  B,)  it  will  divide  it  into  two 

B  equal  parts  of  the  same 
length  and  breadth  as 
the  parallelogram,  but 
containing  only  half  its 
surface.  These  two 
parts  are  triangles. — 

Now    supposing     the 

A  length  of  this  parallelo- 

gram to  be  6  feet,  and  its  breadtlf  2,  the  area  would  be 
12  feet.  But  the  triangle  will  contain  only  half  the  sur- 
face,  or  6  feet. 

Hence  to  find  the  area  of  a  triangle, 
Multiphj  the  length  by  half  the  breadth,  or  tJie  breadth  by 
half  the  length. 

75.  In  a  triangle  32  inches  by  10,  how  many  square 
inches  ?  Ans.  160  sq.  inches. 

76.  What  is  the  area  of  a  triangle  whose  base  is  30 
rods  and  the  perpendicular  6  rods  ?  Ans.  90  rods. 

It  has  been  shown  that  the  length  of  one  side  of  a  cube 
raised  to  its  third  power  will  give  the  solid  contents  of  the 
cube. 

Hence  to  find  the  solid  contents  of  a  cube,  when  one 
side  is  given, 

Muhij^Iy  the  given  side  into  itself  twice,  or  raise  it  to  its^ 
third  fower. 

77.  The  side  of  a  cubic  block  is  12  inches  ;  how  many 
solid  inches  does  the  block  contain  ?  Ans.  123=1728 

7S.  One  side  of  a  cube  is  59  feet :  what  are  its  solid 
contents  ?  '  Ans.  205379 


268  ARITHMETIC.      PART  THIRD. 

79.  If  a  cube  contains  614,125  cubic  yards,  what  is  the 
length  of  one  side  ?  Ans.  85  yards. 

Note.  This  answer  is  obtained  by  finding  the  cube 
root  of  614125. 

A  circle  is  a  figure  contained  by  one  line  called  the  cir. 
eumference,  every  part  of  which  is  equally  distant  from  a 
point  within  called  the  centre. 

The  diameter  of  a  circle,  is  a  line  drawn  through  the 
centre,  dividing  it  into  two  equal  parts. 

It  is  found  by  calculation,  that  the  circuvvference  of  a 
circle  measures  about  3i  times  as  much  as  its  diameter,  or 
more  accurately  in  decimals,  3,4159  times. 

Hence  to  find  the  circumference  of  a  circle  when  the 
diameter  is  known, 

Multiply  the  diameter  by  3|. 

To  find  the  diameter  when  the  circumference  is  known, 

Divide  the  circumference  by  3{. 

To  find  the  area  of  a  circle, 

Midtiply  i  the  diameter  into  ^  the  circumference. 

80.  If  the  diameter  of  a  wheel  is  4  feet,  what  is  its  cir- 
circumference  ?  Ans.  12|  feet. 

81.  What  is  the  circumference  of  a  circle,  whose  di- 
ameter is  147  feet  ?  Ans.  462  feet. 

82.  What  is  the  diameter  of  a  circle,  whose  circum- 
ference is  462  feet  ?  Ans.  147  feet. 

83.  What  is  the  area  of  a  circle,  whose  diameter  is  7 
feet,  and  its  circumference  22  feet  ?       Ans.  38^  sq.  feet. 

84.  What  is  the  area  of  a  circle,  whose  circumference 
is  176  rods?  Ans.  2464  rods. 

The  area  of  a  globe,  or  ball,  is  4  times  as  much  as  the 
area  of  a  circle  of  the  same  diameter. 
Hence,  to  find  the  area  of  a  globe, 
Multiply  the  ivhole  circumference  into  the  whole  diameter. 

85.  What  is  the  number  of  square  miles  on  the  surface 
of  the  earth,  supposing  its  diameter  7911  miles  ? 

Ans.  7911x24353=196,612,083. 
To  find  the  solid  contents  of  a  globe,  or  ball. 
Multiply  its  area  by  J-  part  of  its  diameter. 

86.  How  many  solid  inches  in  a  ball  7  inches  in  diame- 
ter ?  Ans.  179f . 


FORMS  OF  NOTES,  RECEIPTS,  A:C.         269 

A  cylinder  is  a  round  body,  whose  ends  are  circles,  and 
"which  is  of  equal  size  from  end  to  end. 
To  find  the  solid  contents  of  a  cylinder, 
Multiply  the  area  of  one  end  by  the  length. 

87.  There  is  a  cylinder  10  feet  long,  the  area  of  whose 
ends  is  3  square  feet ;  how  many  solid  feet  does  it  con- 
tain  ?  Ans.  30. 

Solids  which  decrease  gradually  from  the  base  till  they 
come  to  a  point,  are  called  pyramids.  The  point  at  the 
top  of  a  pyramid  is  called  the  vertex.  A  line  drawn  from 
the  vertex  perpendicular  to  the  base,  is  called  the  perpen- 
dicular  height  of  the  pyramid. 

To  find  the  solid  contents  of  a  pyramid. 

Multiply  the  area  of  the  base  by  i  of  the  perpendicular 
height. 

88.  There  is  a  pyramid  whose  height  is  9  feet,  and 
whose  base  is  4  feet  square  ;  what  are  its  contents  ? 

Ans.  48  feet. 

89.  There  is  a  pyramid,  whose  height  is  27  feet,  and 
whose  base  is  7  feet  in  diameter  ;  what  are  its  solid  con- 
tents  ?  Ans.  346i  feet. 


FORMS  OF  NOTES,  RECEIPTS,  AND 
ORDERS. 

When  a  man  wishes  to  borrow  money,  after  receiving 
It,  he  gives  his  promise  to  repay  it,  in  such  forms  as  those 
below. 

Notes. 

No.  1. 

Hartford,  Jan.  1,  1S32. 
For  value  received,  I  promise  to  pay  D.  F.  Robinson, 
or  order,  two  hundred  sixty  four  dollars,  twenty-five  cents, 
on  demand,  with  interest.  John  Sjiith. 

No.  2. 

New  York,  Jan.  15,  1832. 
For  value  received,  I  promise  to  pay  William  Dennis, 
or  bearer,  twenty  dollars,  sixteen  cents,  three  months  after 
date.  George  Ellis, 

i>3* 


270  ARITHMETIC.       PART   THIRD. 

No.  3. 

Philadelphia,  July  6,  1831. 
For  value  received,  we,  jointly,  and  severally,  promise 
to   pay  to  Henrj'  Reddy,  or  order,  one  hundred  dollars, 
thirteen  cents,  on  demand,  with  interest. 

James  Barnes. 
Attest.  James  Cook.  Williaim:  Hedge. 

Remarks. 

1.  The  sum  lent,  or  borrowed,  should  be  written  out  in 
words,  instead  of  using  figures. 

2.  When  a  note  has  the  words  "  or  order,"  or  "  or 
hearer,"  it  is  called  negociable  ;  that  is,  it  may  be  given  or 
sold  to  another  man,  and  he  can  collect  it. 

If  the  note  be  written,  to  pay  him  "  or  order,"  (see  No. 
I,)  then  D.  F.  Robinson  can  endorse  the  note,  that  is, 
write  his  name  on  the  back  of  it,  and  then  sell  it  to  any  one 
he  chooses.  Whoever  buys  the  note,  demands  pay  from 
the  signer,  John  Smith. 

3.  If  the  note  be  written,  "  or  hearer,"  (see  note  2,) 
then  whoever  holds  the  note  can  collect  it  of  the  signer. 

4.  When  no  rate  of  interest  is  mentioned,  it  is  to  be 
understood  at  the  legal  rate  in  the  state  where  the  note  is 
given. 

5.  All  notes  are  payable  on  demand,  unless  some  par- 
ticular time  is  specified. 

6.  All  notes  draw  interest  after  the  time  of  promised 
payment  has  elapsed,  even  if  there  is  no  promise  of  inte- 
rest in  the  note. 

7.  Notes  that  are  to  be  paid  on  demand,  draw  interest 
after  a  demand  is  made. 

8.  If  a  man  promises  to  pay  in  certain  other  articles,  in- 
stead of  money,  after  the  time  of  promised  payment  has- 
elapsed,  the  creditor  can  claim  payment  in  money. 

Receipts. 

Hartford,  June  16,  1831. 
Received  of  Mr.  Julius  Feck,  twelve  dollars,  in  full  of 
all  accoun's.  John  Osgood. 


FORMS  OF  NOTES,  KECEIPT3,  6cC.         271 

Receipt  for  money  on  a  note. 

Hartford,  June  18,  1831. 
Received  of  John  Goodman,  (by  the  hand  of  Willliani 
Smith,)   twenty  dollars,  sixteen  cents,  which  is  endorsed 
on  his  note  of  July  6,  1829.  John  Reed. 

Receipt  J  or  money  on  account. 

Hartford,  April  6,  1831. 
Received  of  Albert  Jones,  forty  dollars,  on  account. 

Peter  Trusty. 
Receipt  oj  Money  for  another  Person. 

Hartford,  June  1st,  1831. 
Received  of  A.  B.  one  hundred  and  six  dollars,  for 
I.  C.  Samuel  Wilson. 

Receipt  for  Interest  due  on  a  Note. 

Hartford,  Aug.  1,  1832. 
Received  of  W.    B.    thirty    dollars    in    full  of  one 

year's  interest   of  $500,  due  to  me  on  the day  of 

last,  on  note  from  the  said  W.  B. 

William  Gray. 
Receipt  for  Money  paid  before  it  is  due. 

Newport,  June  1,  1829. 
Received  of  A.  F.  sixty  dollars  advanced,  in  full  for  one 
year's  rent  of  my  house,  leased  to  said  A.  F.  ending  the 
first  day  of  September  next,  1829. 

John  Graves. 

Note. — If  a  receipt  is  given  in  full  of  all  accounts,  it 
cuts  off  only  the  claims  oi  accounts.  But  "  in  full  of  all 
demands^'  cuts  off  all  claims  of  every  kind. 

Orders. 

New  York,  June  9,  1830. 
Mr.  John  Ayers.     For  value  received,  pay  to  N.  S.  or 
order,  fifty  dollars,  and  place  the  same  to  my  account. 

Solomon  Green. 
New  York,  July  9,  1831. 
Mr.  William  Redfield,— Please  to  deliver  Mr.  L.  D. 
such  goods  as  he  may  call  for,  not  exceeding  the  sum  of 
one  hundred  dollars,  and  place  the  same  to  the  account 
of  your  humble  servant.  Stephen  Birch. 


272 


ARITHMETIC.       PART  THIRD* 


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